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Compact Self Adjoint Hilbert Schmidt and Trace Class Operators
1 · Prerequisites
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Banach Valued Integration and the Radon Nikodym Property
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces Adjoint Operators and Annihilators
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Foundations of the Real Numbers for Analysis
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Orthonormal Bases, Parseval and Fourier Series
- Polynomial Rings, the Division Algorithm and Roots
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Square-Integrable Kernels and Hilbert–Schmidt Compactness
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Complex Exponential and Euler's Formula
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
The page begins with the quantitative core of self-adjoint operator theory: the operator norm is the supremum of its quadratic form , proved over both scalar fields by a rotation and rescaling polarisation bound; for a nonzero compact self-adjoint operator that supremum is attained up to sign as an eigenvalue, using only countable choice for the approximate maximisers and the choice-free equivalence of compactness with sequential compactness in metric spaces. Eigenvalues of a self-adjoint operator are real, eigenspaces for distinct eigenvalues are orthogonal, and the orthogonal complement of an eigenspace is again a closed invariant subspace on which the restriction is self-adjoint.
The spectral theorem for compact self-adjoint operators then assembles the theory: nonzero eigenvalues form a finite or countable set of reals, have finite multiplicity, and can accumulate only at ; the closed span of their eigenspaces is ; the operator is the norm limit of its finite spectral partial sums ; and on a complex Hilbert space the nonzero spectrum is exactly the set of nonzero eigenvalues, obtained through an explicit bounded inverse off the eigenvalue set. A Hilbert basis of is never selected, and adjoining one to orthonormal bases of the nonzero eigenspaces gives the orthonormal eigenbasis corollary under full AC. Uniqueness of the compact positive square root of a compact self-adjoint positive operator follows from its spectral expansion, including uniqueness among all compact positive square roots via the symmetric/skew decomposition of a root.
From the square root the page builds the absolute value and the zero-padded singular-value sequence, its singular-value decomposition with orthonormal systems indexed exactly by the positive singular values and the partial isometry satisfying , the identification of singular values with approximation numbers , the resulting compactness criterion compact , and the operator-norm density of finite-rank operators in the compact operators with error the next singular value.
The Hilbert–Schmidt theory is imported from the earlier square-kernel pair and completed here by the two-sided ideal theorem: with bases supplied as data the Hilbert–Schmidt operators form a vector space closed under adjoints, and . The trace-class chain is then developed in its own right: trace class is with trace norm ; products of two Hilbert–Schmidt operators are trace class and every trace-class operator factors through two Hilbert–Schmidt operators attaining the trace norm; the nuclear series , characterises the trace class and computes as the infimum of nuclear sums; trace-class operators form a two-sided Banach ideal. The trace itself is defined first relative to a supplied Hilbert basis by the absolutely convergent sum , then shown to be a single basis-independent scalar equal to for every nuclear representation, using a deterministically constructed separable support Hilbert space rather than a basis of the ambient space; cyclicity follows by rank-one computation and trace-norm density, and for self-adjoint positive operators the trace is the eigenvalue sum .
The final draft block conditionally extends this to arbitrary complex Hilbert spaces: a source-backed external separable determinant theorem is recorded with its precise algebraic-multiplicity and zero-space conventions, the local definition extends it through a separable reducing support, and the determinant properties yield general nonnormal Lidskii. These records prominently retain their external-proof status; the future determinant module must replace the external theorem rather than create duplicate determinant or Lidskii items.
3 · Logical flowchart
4 · Definitions, theorems and proofs
Norm of a self adjoint operator from its quadratic form
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space) and let be a bounded self-adjoint operator (Self-adjoint, positive, unitary and normal operators, A bounded linear operator between normed spaces), with operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum). Put for and
with the convention that a supremum over the empty set of reals is ; the empty case occurs only for , where the only operator is . Then
The identity holds over both scalar fields, and in the case both sides equal .
Facts & Assumptions
Given: A real or complex Hilbert space , a bounded self-adjoint operator , the quadratic form , and with the empty-supremum convention.
Self-adjointness is the identity for all : a self-adjoint operator satisfies , and the Hilbert adjoint is characterised by (Self-adjoint, positive, unitary and normal operators, The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities).
Inner-product algebra. The pairing is linear in the first argument, conjugate-linear in the second, conjugate symmetric, and positive definite, and (Real and complex inner-product spaces and their induced length). Consequently for real and one has , the expansion holds, and if is real then self-adjointness gives , so that . For with unit vector one has .
Parallelogram law. for all (Pythagoras, the parallelogram identity, and the real and complex polarisation identities).
Operator norm and Cauchy–Schwarz. and (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces); (Cauchy–Schwarz: , with equality exactly for dependent pairs), so the dual norm formula holds for every by Cauchy–Schwarz and by testing when (both sides are at ).
Choice. Countable Choice is the hypothesis under which this pair's Hilbert-space interface is stated; no choice is used inside the argument below (The Axiom of Countable Choice ()).
Proof
Given: Countable Choice, a real or complex Hilbert space , a bounded self-adjoint , and .
The quadratic form is dominated by . For , if then , while if then [A2] gives for the unit vector , hence .
. For every unit vector , Cauchy–Schwarz and the norm bound give , so is an upper bound of the set whose supremum is ; hence , and when both numbers are .
A sesquilinear bound. For all one has : if or then and the right side is ; otherwise put , and if choose the unit scalar with (namely over , over ) and set , so that and is real; then [A2] gives by [step 1.1], and the parallelogram law [A3] turns the last factor into .
Removing the norms. For and every real , [step 2.1] applied to the pair together with the scaling identity of [A2] gives ; the right side is minimised at , where it equals , so for all (the zero cases being trivial).
. For the dual norm formula [A4] gives by [step 3.1] with , and the inequality also holds at ; thus is a uniform bound for on the unit ball, so by the unit-ball characterisation of the operator norm in [A4].
Conclusion. Steps 1.2 and 4.1 give ; if then , by the empty-supremum convention and , so the identity holds there as well.
Norm point of a compact self adjoint operator is an eigenvalue up to sign
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space) and let be a nonzero compact self-adjoint operator (Compact linear operator, Self-adjoint, positive, unitary and normal operators, A bounded linear operator between normed spaces). Then either or is an eigenvalue of (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism) and possesses a unit eigenvector; here .
Facts & Assumptions
Given: Countable Choice, a nonzero compact self-adjoint operator on a Hilbert space , and the quadratic form .
Norm formula and positivity of the norm. with the empty-supremum convention; forces and , and for every real there is a unit vector with (Norm of a self adjoint operator from its quadratic form, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
Self-adjointness. is real-valued and for all , so for every (Self-adjoint, positive, unitary and normal operators, The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities).
Compactness and ZF metrisation. compactness of is tested on the closed unit ball: for compact the set is a compact subset of , and conversely a compact closure of that image forces to be compact, where is the closed unit ball; a compact metric space is sequentially compact, and that implication is a theorem of ZF (Compact linear operator, In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle).
Continuity, norms and limits. Bounded linear operators are continuous and satisfy (A bounded linear operator between normed spaces, For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent, The operator norm as the least bound and as the unit-sphere or unit-ball supremum); limits of sequences in a metric space are unique, convergence of norms gives , and a continuous map carries convergent sequences to convergent sequences (Convergence of a sequence in a metric space: iff in , A sequence in a metric space has at most one limit).
Choice and enumeration. Countable Choice supplies one unit vector for each from the nonempty set ; the increasing enumeration of an infinite subset of is defined by recursion and is choice-free (The Axiom of Countable Choice (), The recursion theorem, The well-ordering principle).
Eigenvalues. A scalar is an eigenvalue of when for some , and such an is a unit eigenvector when in addition (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism).
Proof
Given: Countable Choice, a nonzero compact self-adjoint , its quadratic form , and the closed unit ball .
Approximate maximisers. For each , the number is strictly below the supremum in [A1], so the set is nonempty. Countable Choice [A5] supplies a function with for every . Thus is a zero-based sequence of unit vectors with the required bound.
A constant sign on a subsequence. Put and . Since the infinite set is the union of and , at least one of those two sets is infinite. If is infinite let be its increasing enumeration and set ; otherwise let enumerate the infinite set and set . In either case every , so is defined. Then and for every , because has the sign of on this subsequence and , and .
A convergent image subsequence. The set is compact by compactness of [A3], and for every because by [step 1.1], so by sequential compactness of the compact metric space [A3] there are a strictly increasing sequence and a point with .
The residual tends to zero. For every , using [A2], and [step 2.1], , the inequality using ; since and is fixed, .
The approximating vectors converge. For each the identity holds because (indeed by [A1]); the first term converges to by [step 2.2] and the second to by [step 3.1], so .
Conclusion. By continuity of the norm and we get [A4], and by continuity of and uniqueness of limits while also , so ; thus is an eigenvalue of with the unit eigenvector .
Eigenspaces of a self adjoint operator are orthogonal
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space) and let be self-adjoint (Self-adjoint, positive, unitary and normal operators). Then:
- every eigenvalue of (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism) is a real number: if with , then ;
- eigenspaces belonging to distinct eigenvalues are orthogonal: if and , , then (Orthogonality and the orthogonal complement).
Facts & Assumptions
Given: A real or complex Hilbert space and a self-adjoint bounded operator on .
Self-adjointness. , so for all (Self-adjoint, positive, unitary and normal operators, The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities).
Inner-product algebra. The pairing is linear in the first argument, conjugate-linear in the second, conjugate symmetric and positive definite, and (Real and complex inner-product spaces and their induced length); in particular , so this number is its own conjugate and lies in .
Eigen-data. means , so is an eigenvalue with eigenvector for the nonzero members of that kernel, that is , and then by positive definiteness (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism, Real and complex inner-product spaces and their induced length).
Scalars. For one has only for , and conjugation fixes every real scalar; if is real and , then (Real and complex inner-product spaces and their induced length).
Countable Choice is the standing hypothesis of this pair's Hilbert-space interface (The Axiom of Countable Choice ()).
Proof
Given: Countable Choice, a self-adjoint , and eigen-data as in the statement.
Eigenvalues are real. Let with . By conjugate symmetry, [A1] applied to the pair and conjugate-linearity in the second argument, ; since is a nonzero real number, [A4] gives , that is .
Distinct eigenvalues force orthogonality. Let and with and . By [A1] and conjugate-linearity in the second argument, ; by [step 1.1] both and are real, so and hence ; since , it follows that .
Conclusion. Claim 1 is [step 1.1]. For claim 2 let and with : if both are nonzero then [step 2.1] gives , while if or then as well because the pairing is additive and homogeneous in the first argument and conjugate-linear in the second (Real and complex inner-product spaces and their induced length).
Orthogonal complement of an eigenspace is invariant
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space), let be self-adjoint (Self-adjoint, positive, unitary and normal operators) and let be an eigenvalue of (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism) with eigenspace . Then:
- is a closed linear subspace of and ;
- (Orthogonality and the orthogonal complement) is a closed linear subspace of and ;
- the restrictions and satisfy the self-adjoint identity for all in the respective subspace.
Facts & Assumptions
Given: A Hilbert space , a self-adjoint bounded , an eigenvalue , and .
Eigenspace and self-adjointness. is a linear subspace of (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism, Kernel and image of a linear map, Linear subspace of a vector space); is self-adjoint, so for all (Self-adjoint, positive, unitary and normal operators, The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities).
Continuity and limits. The bounded operator is continuous, so implies , and limits of convergent sequences in a metric space are unique (A bounded linear operator between normed spaces, For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent, Convergence of a sequence in a metric space: iff in , A sequence in a metric space has at most one limit).
Complements and closedness. For every subset of an inner-product space the orthogonal complement is a closed linear subspace (Orthogonal complements are closed, Orthogonality and the orthogonal complement); means for every , and the pairing is linear in the first argument and conjugate-linear in the second (Real and complex inner-product spaces and their induced length).
Countable Choice is the standing hypothesis of this pair's Hilbert-space interface (The Axiom of Countable Choice ()).
Proof
Given: Countable Choice and the data above.
is a closed linear subspace. Let with ; then for all , so by continuity [A2] both and , and uniqueness of limits gives , i.e. ; with the linear-subspace statement of [A1] this proves closedness.
is -invariant. If then by [A1] and linearity of the subspace.
is closed. The general closedness of orthogonal complements [A3] applied to the subset gives that is a closed linear subspace of .
is -invariant. Let and . By self-adjointness and , , since by definition of the orthogonal complement; as was arbitrary, for all , that is .
The restrictions are self-adjoint. If both lie in , or both lie in , then and [A1] gives in , which is exactly the defining identity of self-adjointness for the restricted operator on that subspace; [step 1.2] and [step 1.4] show that each restriction maps its subspace into itself, and [step 1.1] and [step 1.3] give the closedness statements of claims 1 and 2.
Spectral theorem for compact self adjoint operators
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space) and let be a compact self-adjoint operator (Compact linear operator, Self-adjoint, positive, unitary and normal operators, A bounded linear operator between normed spaces). Let
be its set of nonzero eigenvalues (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism), with eigenspaces for . Then:
- is a finite or countably infinite set of real numbers, each eigenvalue has finite multiplicity in the sense , and for every real there are only finitely many with ; in particular every point of has a neighbourhood containing only finitely many elements of , so the only possible accumulation point of is ;
- the closed linear span of satisfies (Orthogonality and the orthogonal complement), and with ;
- for every the finite-subset net of over the orthogonal projections onto converges in norm and
- if in addition is a complex Hilbert space, then the nonzero spectrum agrees with the nonzero eigenvalues,
No Hilbert basis of is selected anywhere: only the orthonormal bases of the finite-dimensional eigenspaces , , are used.
Facts & Assumptions
Given: Countable Choice, a real or complex Hilbert space , a compact self-adjoint , the set of nonzero eigenvalues, their eigenspaces , and (the closed linear span).
Self-adjointness and eigenspaces. is real and for all ; is the eigenspace of and is a linear subspace (Self-adjoint, positive, unitary and normal operators, The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities, Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism, Linear subspace of a vector space, Kernel and image of a linear map).
Extremal eigenvalue and orthogonality. Every nonzero compact self-adjoint operator has or as an eigenvalue with a unit eigenvector (Norm point of a compact self adjoint operator is an eigenvalue up to sign); eigenvalues of a self-adjoint operator are real and distinct eigenspaces are orthogonal (Eigenspaces of a self adjoint operator are orthogonal); and are closed -invariant subspaces on which satisfies the self-adjoint identity (Orthogonal complement of an eigenspace is invariant).
Compactness and closed subspaces. is compact exactly when is a compact subset of , where (Compact linear operator, Open ball, closed ball and sphere in a metric space); scalar multiples and continuous images of compact sets are compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism); a closed subset of a compact space is compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact); a compact metric space is sequentially compact, choice-free (In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle); a closed subspace of a complete metric space is complete, in ZF (Closed subspaces of complete metric spaces are complete; the converse under countable choice); is a Banach space (Banach space, A bounded linear operator between normed spaces).
Subspace compactness. If is a closed subspace of , the inclusion is a bounded linear operator and is compact (Compositions with a compact operator are compact, A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
Finite dimension. A normed space has compact closed unit ball exactly when it admits an ordered basis of finite length (The closed unit ball is compact if and only if the normed space is finite-dimensional); every finite-dimensional real or complex inner product space has an orthonormal basis, the empty one in dimension zero (Every finite-dimensional real or complex inner product space has an orthonormal basis); an orthonormal family is linearly independent with unit vectors (Orthonormal families, complete orthonormal systems and Hilbert bases).
Orthogonal complements and expansion. For every subset , is a closed linear subspace; means (Orthogonality and the orthogonal complement, Orthogonal complements are closed); for a linear subspace (The double orthogonal complement of a subspace is its closure); for closed , with unique decomposition (Orthogonal decomposition by a closed subspace); the finite-subset net of converges to for a complete orthonormal family, with Parseval's identity (Fourier expansion in a Hilbert space, Parseval equivalences for an orthonormal family); finite Bessel: (The finite Bessel inequality and best approximation by a finite orthonormal family); a square-summable orthogonal family has a norm-convergent finite-subset net whose limit has the sums of the squared norms (Square-summable orthogonal families have norm-convergent finite sums, Square-summable families on an arbitrary index set and the space ).
Cardinality and Archimedes. Under a countable union of at most countable sets is at most countable, and subsets of at most countable sets are at most countable (Countable unions of at most countable sets, assuming , Every subset of an at most countable set is at most countable, Finite, countably infinite, countable, uncountable, The Axiom of Countable Choice ()); for every real there is a natural with (For every in a complete ordered field there is a natural with ).
Continuity, limits, scalars. Bounded linear operators are continuous with (A bounded linear operator between normed spaces, For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent, The operator norm as the least bound and as the unit-sphere or unit-ball supremum); limits of sequences are unique (Convergence of a sequence in a metric space: iff in , A sequence in a metric space has at most one limit); the complex distance is (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane); the inner product is additive and homogeneous in the first argument (Real and complex inner-product spaces and their induced length).
Spectrum. For a complex Banach space, means is bijective with bounded inverse, , and every eigenvalue lies in (Spectrum and resolvent of a bounded operator).
Proof
Given: Countable Choice, the compact self-adjoint , the set of its nonzero eigenvalues, the eigenspaces , the closed span , and the closed unit ball .
Nonzero eigenspaces are finite-dimensional. Let and let , which is compact by [A3]. If with then , so ; thus , and is compact as a continuous image of a compact set [A3]. The set is closed [A2], so is closed in and hence compact as a closed subset of the compact set [A3]; by the closed-unit-ball criterion [A5] applied to the normed space , the space has finite dimension, so its multiplicity is finite.
Only finitely many eigenvalues above each threshold. Fix and put . The compact set has a finite cover by open balls of radius with centres in , by applying compactness to the cover by all such balls [A3]. For each ball in that finite cover let consist of those for which for some unit . If distinct belonged to , their witnessing unit eigenvectors would be orthogonal by [A2], and hence , whereas two points of have distance less than . Thus each has at most one element. Every has a unit eigenvector whose image belongs to , so is finite. This argument makes no infinite choice of eigenvectors. For , the ball meets only inside , proving local finiteness away from zero.
is annihilated by . Since is the closed linear span of the subspaces , a vector is orthogonal to exactly when it is orthogonal to every ; hence is a closed linear subspace, and it is -invariant because each is -invariant [A2]. As a closed subspace of the Hilbert space , is complete [A3], and the restriction is compact: [A4] gives compact closure in of each bounded image, that closure lies in the closed subspace , and its subspace topology is unchanged and self-adjoint, since for one has in and both vectors lie again in . If , the extremal eigenvalue lemma [A2] provides and with , hence and ; then , so and by positive definiteness, a contradiction. Therefore , that is vanishes on .
An orthonormal family with closed span . By [step 1.2] each set , , is finite, and every lies in some because and for a suitable by [A7]; hence is at most countable by [A7]. By Countable Choice [A7], choose for every an orthonormal basis of the finite-dimensional space , which exists by [A5], and let be the disjoint union of these finite families indexed by the at most countable set , so that is at most countable. Every has norm , and orthonormal bases of orthogonal eigenspaces [A2] make an orthonormal family whose closed linear span is by the definition of .
The support of the operator. Every eigenvector with nonzero eigenvalue is orthogonal to , because for and with one has by self-adjointness, so ; since is closed [A6] and contains each , it contains , while [step 1.3] and [A6] give ; hence . Moreover by the same computation read with arbitrary, so , and if then for every , whence and ; applying [A6] to the linear subspace gives , while the previous inclusion reverses after taking complements: .
The spectral expansion. Let . By [A6] and [step 2.2] there is a unique decomposition with and , and by [step 1.3] , so . By [step 2.1] the family is complete in , so the Fourier expansion [A6] gives as the limit of the finite-subset net, and the same holds with in place of because . Writing , the family is orthogonal with by Bessel [A6], so by [A6] the finite-subset net converges to some ; for every the difference is orthogonal to , hence to , so and . Finally for finite because each , and continuity of [A8] carries the convergent net to ; hence the finite-subset net of converges to , and adding gives .
A spectral gap off the eigenvalue set. Let with . The set is finite by [step 1.2], and ; set , a positive real number because is finite and every displayed distance is positive. For every one has : if this is the definition of , and if then by the triangle inequality [A8]. Consequently, for every the orthogonal family has by Bessel [A6], so [A6] makes its finite-subset net converge to a vector of norm .
The candidate inverse. Fix with and, for , write with , as in [step 3.1]. The limit of the finite-subset net in [step 3.2] is unique: if the same net converges to and to , then for any there are finite sets such that every has and every has ; at the triangle inequality gives , and hence . We may therefore define
where the first term is that unique limit. The map is linear because each is linear and limits respect linear combinations, and by orthogonality of the decomposition [A6], so is a bounded linear operator on .
The inverse identities and the spectrum. Work now over and fix outside . For finite , put . Since acts as on , one has . By [step 3.2] and continuity of , taking limits gives . Also because . Thus . For the other identity, self-adjointness and the finite formula for give : for each basis vector , , since is real. Moreover by [step 2.2], so by uniqueness of the orthogonal decomposition. Consequently . This proves that is a bounded two-sided inverse, so [A9]. Conversely a nonzero eigenvector makes noninjective, so every lies in . Therefore .
Conclusion. Claim 1 is the combination of [step 1.1], [step 1.2], [step 2.1] and the reality of eigenvalues in [A2]; claim 2 is [step 2.2] together with the decomposition of [step 3.1]; claim 3 is [step 3.1]; claim 4 is [step 5.1]. At no point was a Hilbert basis of selected: the chosen vectors all lie in the eigenspaces with , which are contained in by [step 3.1].
Orthonormal eigenbasis for a compact self adjoint operator
Statement
Assume the Axiom of Choice (The Axiom of Choice). Let be a real or complex Hilbert space (Hilbert space) and let be a compact self-adjoint operator (Compact linear operator, Self-adjoint, positive, unitary and normal operators). Then has a Hilbert basis (Orthonormal families, complete orthonormal systems and Hilbert bases) consisting of eigenvectors of (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism): one may take the union of orthonormal bases of the finitely many-dimensional nonzero eigenspaces with a Hilbert basis of . If no vectors from the kernel are needed, and if all nonzero eigenspaces are absent (that is ) the union is a Hilbert basis of .
Facts & Assumptions
Given: AC, a real or complex Hilbert space , a compact self-adjoint , the set of its nonzero eigenvalues, the eigenspaces for , and .
Spectral theorem. is finite or countably infinite with finite multiplicities, the nonzero eigenvalues are real and distinct eigenspaces are orthogonal, and with (Spectral theorem for compact self adjoint operators, Eigenspaces of a self adjoint operator are orthogonal, Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism).
Orthonormal families. An orthonormal family has unit vectors which are pairwise orthogonal, and every finite-dimensional real or complex inner product space has an orthonormal basis, the empty one in dimension zero; the closed linear span of an orthonormal family is a closed subspace, and a Hilbert basis is a complete orthonormal family (Orthonormal families, complete orthonormal systems and Hilbert bases, Every finite-dimensional real or complex inner product space has an orthonormal basis).
Complements. is a closed linear subspace for every subset ; orthogonality is symmetric and bilinear in the obvious sense; for closed one has with (Orthogonality and the orthogonal complement, Orthogonal complements are closed, Orthogonal decomposition by a closed subspace).
Zorn and AC data. Under AC every nonempty poset in which every chain has an upper bound has a maximal element (Zorn's lemma, The Axiom of Choice, Partial order and partially ordered set, Chain in a poset, Upper bound, least upper bound, and strict upper bound, Maximal element and greatest element); AC implies (AC supplies the countable and dependent choices used in Banach integration, The Axiom of Countable Choice ()).
Closed subspaces are complete. A closed subspace of a complete metric space is complete in ZF, so a closed subspace of a Hilbert space is again a Hilbert space (Closed subspaces of complete metric spaces are complete; the converse under countable choice, Hilbert space). The set of orthonormal families in a subset of is a poset under inclusion, and the inclusion-union of a chain of orthonormal families is orthonormal (Orthonormal families, complete orthonormal systems and Hilbert bases, Partial order and partially ordered set, Chain in a poset).
Proof
Given: AC, the compact self-adjoint , the nonzero eigenspaces , their closed span , and the kernel .
An orthonormal family spanning . For every the eigenspace is finite-dimensional by [A1], so it has an orthonormal basis by [A2]; the disjoint union of all these finite bases is an orthonormal family, because each basis is orthonormal and vectors belonging to distinct eigenvalues are orthogonal by [A1]. Its closed linear span is by the definition of , and by [A1].
A maximal orthonormal family in the kernel. Let be the set of orthonormal families contained in , ordered by inclusion. This is a nonempty poset (the empty family belongs to it) and the union of any chain in is again an orthonormal family contained in , hence an upper bound of the chain; therefore Zorn's lemma [A4] provides a maximal element .
is complete in the kernel. Let be the closed span of , which is a closed subspace of the Hilbert space [A5]; if then by the orthogonal decomposition in the Hilbert space [A3] there is with . Then has norm , is orthogonal to every element of , and lies in , so is an orthonormal family in strictly containing , contradicting maximality; hence , that is is a complete orthonormal family of the Hilbert space .
The union is a Hilbert basis of . The union is an orthonormal family: it is the union of two orthonormal families, and every is orthogonal to every by [step 1.1] and [A1]. Its closed linear span contains (by [step 1.1]) and (by [step 2.1]), hence contains by [A1]; therefore is complete and is a Hilbert basis of .
Conclusion. Every element of is an eigenvector of : the vectors of lie in nonzero eigenspaces, while every is a unit vector in , so and (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism). Thus, whenever is nonempty, is an eigenvalue witnessed by each of its members; when one has , and the proof neither needs nor asserts that is an eigenvalue. By [step 3.1] the family is a Hilbert basis of consisting of eigenvectors of , which proves the corollary; the degenerate descriptions in the statement are the cases (then and spans ) and (then ).
Positive square root of a compact positive operator
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space) and let be a compact self-adjoint positive operator (Compact linear operator, Self-adjoint, positive, unitary and normal operators, A bounded linear operator between normed spaces), so that is a nonnegative real for every . Then there is a compact self-adjoint positive operator with , and it is unique: if is compact and positive (Self-adjoint, positive, unitary and normal operators) with , then . The root acts by multiplication by on each positive eigenspace , , and by zero on ; in particular is the operator denoted or .
Facts & Assumptions
Given: Countable Choice, a real or complex Hilbert space , a compact self-adjoint positive , the set is an eigenvalue of , the eigenspaces , and .
Spectral theorem for . is finite or countably infinite, each has finite dimension and an orthonormal basis, distinct eigenspaces are orthogonal, , and with in norm for , where is the orthogonal projection onto (Spectral theorem for compact self adjoint operators, Eigenspaces of a self adjoint operator are orthogonal, Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism, Orthonormal families, complete orthonormal systems and Hilbert bases).
Stability data. The are finite sums of rank-one maps and satisfy and ; the family is orthogonal with (Bessel) and the expansion of over the union of orthonormal bases of the converges to the component of in (The finite Bessel inequality and best approximation by a finite orthonormal family, Parseval equivalences for an orthonormal family, Fourier expansion in a Hilbert space, Orthogonality and the orthogonal complement, Real and complex inner-product spaces and their induced length).
Square-summable orthogonal families. If is an orthogonal family in a Hilbert space with , then the finite-subset net of converges and the limit satisfies (Square-summable orthogonal families have norm-convergent finite sums, Square-summable families on an arbitrary index set and the space ).
Orthogonal decomposition. For a closed subspace one has with closed and direct, and a bounded linear operator that vanishes on and on is zero; limits of convergent sequences are unique (Orthogonal decomposition by a closed subspace, Linear subspace of a vector space, Convergence of a sequence in a metric space: iff in , Hilbert space, Banach space).
Compactness and finite rank. A finite-rank bounded operator is compact; under a norm limit of compact operators into a Banach space is compact; scalar multiples and images of compact sets under continuous maps are compact (Bounded finite rank operators are compact, Norm limit of compact operators is compact, Compact linear operator, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent).
Finite spectral thresholds. For each real there are only finitely many with ; in particular is finite for every (Spectral theorem for compact self adjoint operators).
Proof
Given: Countable Choice, the compact self-adjoint positive , its positive eigenvalues , the eigenspaces and the projections , and .
Construction of the root. Positivity forces for every eigenvalue of , because for a unit eigenvector one has ; hence consists of positive numbers and is an eigenvalue only in the form of the kernel [A1]. For set , a finite-subset net over the at most countable index set . The family is orthogonal with , using the expansion, orthogonality and Bessel [A1, A2]; so [A3] makes the net converge, is well defined with , and is linear with . Moreover , because is self-adjoint and [A2]; and is self-adjoint, since by absolute convergence and self-adjointness of each . Finally : on one has whence , both operators are continuous and agree on the linear span of , hence on by continuity, and both vanish on (for , for all because and is in every ), so on by [A4].
Every positive square root kills the kernel. Let be compact and positive with , and let . Put , so . For every real , positivity at gives . Here is real and nonnegative. If , choosing makes the right side , impossible. Thus for every . This works over both scalar fields and uses neither self-adjointness nor compactness of .
The root is compact. For each let , finite by [A7], and put . This operator has finite-dimensional range by [A1], so is compact by [A5], including when is empty. For every , the orthogonal summation identity [A3] and Bessel [A2] give . Thus . Since is Banach, [A5] implies that is compact. The same zero-based sequence handles empty, finite and infinite .
A positive square root acts diagonally. Let be as in [step 1.2], let , and put . For set . The identity gives . Positivity therefore implies , forcing . Hence on the entire eigenspace, without a diagonalization of or an assumption that is self-adjoint.
Existence. By [step 1.1] and [step 1.3] the operator is a compact self-adjoint positive operator with , and by construction it acts as on each , , and as on .
Uniqueness. Let be compact and positive with . By [step 1.2] vanishes on , by [step 2.1] it equals on each , and by [step 2.2] the same two descriptions hold for ; hence the bounded operator vanishes on and on each . Since is the closed linear span of by [A1], continuity gives .
Conclusion. The operator of [step 1.1] is compact, self-adjoint and positive with by [step 2.2], and [step 3.1] shows that every compact positive with equals ; the action of on the positive eigenspaces and on the kernel is stated in [step 2.2]. This proves the lemma, including the uniqueness among compact positive square roots.
Absolute value and singular values of a compact operator
Definition
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be real or complex Hilbert spaces (Hilbert space) and let be a compact operator (Compact linear operator, A bounded linear operator between normed spaces), with Hilbert adjoint (The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities).
The absolute value. The operator is compact, since it is the composite of the compact with the bounded (Compositions with a compact operator are compact); it is self-adjoint, because (Hilbert-adjoint identities); and it is positive, because for every (Self-adjoint, positive, unitary and normal operators). The absolute value of is the unique compact self-adjoint positive operator with , whose existence and uniqueness are the preceding square-root lemma (Positive square root of a compact positive operator). It satisfies so in particular (The operator norm as the least bound and as the unit-sphere or unit-ball supremum) and , since for every .
The singular values. By the spectral theorem for (Spectral theorem for compact self adjoint operators) the nonzero eigenvalues of form a finite or countably infinite set of positive reals, each with finite multiplicity, and for every real only finitely many of them exceed ; positivity rules out negative eigenvalues and already corresponds to the kernel. The multiset of positive singular values of is the multiset of positive eigenvalues of , counted with multiplicity (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism).
The ordered singular-value sequence. The distinct positive singular values are listed with positive labels in decreasing order as follows: if the multiset of positive eigenvalues is empty (equivalently , equivalently ), the list is empty; otherwise is an eigenvalue of , a maximum and not merely a supremum, because a supremum value not attained would be an accumulation point different from ; having chosen , put is an eigenvalue of and whenever that set is nonempty, and stop otherwise. Each step is legitimate by the finiteness-above-thresholds property above, and an infinite list satisfies (otherwise its decreasing limit would be a nonzero accumulation point). Writing for the multiplicity of (a positive integer), the zero-padded singular-value sequence is where , , and so on; if the multiset is finite with total multiplicity , one sets for every , and if one sets for every . The number is written and called the -th singular value of .
Zero-based domain and positive labels. Set . Thus the numerical sequence is the function on all of , including zero. The positive-labelled tail is the multiplicity-counting list constructed above. The auxiliary initial value is not an additional entry of the eigenvalue multiset, does not index a singular vector, and is excluded from multiplicity counts and singular-value sums, which use . The full sequence satisfies and tends to zero. This preserves the page's positive rank labels while giving convergence statements a zero-based domain.
Rank and the finiteness of the list. The map given by is well defined and linear, because forces and hence ; it is injective, because gives and ; it is surjective onto because ; and it is isometric, . Hence is finite-dimensional if and only if is finite-dimensional; in that case the linear bijection gives (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis). Consequently, whenever has finite rank, the positive singular values with multiplicity number exactly , the rank of , and all later vanish, so the sequence is zero-padded. If does not have finite rank, the multiset of positive singular values is infinite and countable (Finite, countably infinite, countable, uncountable) and for every , with ; in particular finite rank of is characterised by the eventual vanishing for all sufficiently large , and conversely such eventual vanishing forces finite rank. The sequence is numerical data only: no orthonormal system is selected here, and the zero padding is not an indexing of any family of vectors. The unordered multiset determines uniquely, so is well defined, and .
Singular value decomposition for compact operators
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be real or complex Hilbert spaces (Hilbert space), let be a compact operator (Compact linear operator), let and the singular values be as in the absolute-value definition (Absolute value and singular values of a compact operator). Let when is infinite-dimensional. When is finite-dimensional, put (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis) and let , interpreted as when . Thus indexes exactly the positive singular values counted with multiplicity, which we write as in nonincreasing order. Then:
- there are orthonormal families in and in , indexed by exactly , with and for every ;
- for every the series converges in norm and and its finite partial sums satisfy for every with (and for when );
- the linear map defined on the span of by , extended by continuity to and by zero on , is a partial isometry with and the orthogonal projection onto ;
- the zero-padded sequence is not used to index the orthonormal systems: the systems carry exactly the index set of the positive singular values, and the terms beyond the rank in the finite-rank case are numerical padding only.
Facts & Assumptions
Given: Countable Choice, compact , its absolute value , the index set of the positive singular values with multiplicity, the finite dimension when the range is finite-dimensional, and the zero-padded sequence .
Absolute value and finite rank. is compact, self-adjoint and positive with , and ; the positive singular values with multiplicity are the positive eigenvalues of with multiplicity. They are finite in number exactly when is finite-dimensional, and otherwise form a countably infinite list. In the finite-dimensional case the isometric linear bijection , , gives (Absolute value and singular values of a compact operator, Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis). No value is used.
Spectral theorem for . The nonzero eigenvalues of are positive, have finite-dimensional eigenspaces , are mutually orthogonal across distinct , and their closed span is ; moreover and , so the closed span of the eigenspaces is (Spectral theorem for compact self adjoint operators, Eigenspaces of a self adjoint operator are orthogonal, Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism, Orthogonality and the orthogonal complement, Orthogonal decomposition by a closed subspace).
Bases and expansion. Every finite-dimensional eigenspace has an orthonormal basis (Every finite-dimensional real or complex inner product space has an orthonormal basis); an orthonormal family is complete in a closed subspace in the case, and only in the case, that the finite-subset net of Fourier sums converges there, with Parseval and Bessel inequalities available (Fourier expansion in a Hilbert space, Parseval equivalences for an orthonormal family, The finite Bessel inequality and best approximation by a finite orthonormal family, Orthonormal families, complete orthonormal systems and Hilbert bases).
Continuity. Bounded operators are continuous and satisfy ; limits are unique (A bounded linear operator between normed spaces, For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Convergence of a sequence in a metric space: iff in ).
Countable Choice supplies, for the at most countable eigenvalue list, one orthonormal basis of each finite-dimensional eigenspace (The Axiom of Countable Choice (), Finite, countably infinite, countable, uncountable).
Proof
Given: Countable Choice, the compact , its absolute value , the index set and the singular values , the finite integer when is finite-dimensional, and the eigenspaces of for positive eigenvalues .
Choosing the left system. By [A2] the positive eigenvalues of are precisely the positive singular values with multiplicity, and their eigenspaces are finite-dimensional with closed span ; listing those eigenvalues with multiplicity as and choosing by [A5] an orthonormal basis of each gives an orthonormal family with for every , whose closed linear span is , and the terms are in nonincreasing order.
The right system is orthonormal. For put , which lies in and is well defined because . For , using and the eigenvector property of [step 1.1], so is orthonormal.
The expansion. Let . By [A2] write with and . Since is complete in [step 1.1], the Fourier expansion [A3] gives as a norm limit of finite-subset partial sums, and then continuity of [A4] gives , because and the image net of the finite partial sums converges. Moreover for finite the remainder is whenever , by orthonormality [step 2.1] and Bessel [A3], so the partial sums of the statement satisfy for , and for in the finite-rank case because then for and every index in is .
The right system spans the range closure. Each lies in by [step 2.1], so the closed linear span is contained in ; conversely [step 3.1] exhibits every as the norm limit of finite linear combinations of the , so and hence .
The partial isometry and . Define first on the linear span of by for finite . This is well defined because is linearly independent as an orthonormal family, and it is isometric, since by [step 2.1] ; by [step 1.1] the closure of is , so extends uniquely to a bounded linear operator, still denoted , on with for all and by [step 4.1]. Extend to by on ; then is bounded and, because is self-adjoint with , for every and on [A1], so by continuity on the closed span of and the , which is by [A2]. Finally and : for one has where is the orthogonal projection onto , because is isometric on and vanishes on , so and ; dually, for the vector is characterised by for all , so for and for , that is is the orthogonal projection onto .
Conclusion. Claim 1 is [step 1.1] and [step 2.1]; claim 2 is [step 3.1], whose index set is by construction; claim 3 is [step 5.1] together with [step 4.1]. Claim 4 is the indexing discipline used throughout: indexes the positive singular values with multiplicity and is only for , when the finite dimension is ; it is finite exactly when the range is finite-dimensional. In that case the vanishing terms with are numerical padding and index no vector.
Singular values equal approximation numbers
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be Hilbert spaces over the same field , let be compact (Compact linear operator, A bounded linear operator between normed spaces) and let be its zero-padded singular-value sequence (Absolute value and singular values of a compact operator). For put (Greatest lower bound (infimum), The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis). The displayed set is nonempty because it contains , and it is bounded below by ; its infimum therefore exists by the real infimum property (Every nonempty set bounded below has an infimum). Then including the zero-padded case: if has finite rank and then both numbers are , and if both are for every .
Facts & Assumptions
Given: Countable Choice, a compact , its singular system , , from the singular-value decomposition, and the numbers .
Singular-value decomposition. With the index set of the positive singular values with multiplicity, there are orthonormal systems and with and , the expansion holds in norm, and for every with the partial sum satisfies ; moreover for all when , and in that case (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator).
Ranks of truncations. A finite sum over a finite has range contained in the span of the finitely many , hence rank at most ; for the truncation therefore has rank (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis, Kernel and image of a linear map).
Rank–nullity in finite dimensions. A linear map of a finite-dimensional space satisfies ; hence a linear map on a finite-dimensional space of dimension with rank has a nonzero kernel (Rank-nullity: , Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis, Kernel and image of a linear map).
Orthonormal expansion in the span. If lies in the span of finitely many members of the orthonormal family , then , for and for ; in particular, for in the span of the expansion of reduces to the finite sum ; the finite Bessel inequality bounds partial sums of coefficients (Orthonormal families, complete orthonormal systems and Hilbert bases, The finite Bessel inequality and best approximation by a finite orthonormal family, Real and complex inner-product spaces and their induced length).
Infimum. Every nonempty lower-bounded subset of has an infimum (Every nonempty set bounded below has an infimum); by the defining greatest-lower-bound property, every lower bound of satisfies , and conversely makes a lower bound of (Greatest lower bound (infimum)).
Countable Choice is the standing hypothesis of this pair's Hilbert-space interface (The Axiom of Countable Choice ()).
Proof
Given: Countable Choice, the compact , its singular system and the numbers .
Upper bound . For the truncation (empty for ) has rank by [A2], so . If then [A1] with in place of gives ; if then , and by [A1], so . Finally if then . In every case .
Lower bound . Let have finite-dimensional range with . If then and there is nothing to prove; assume therefore , so that exist and their span has dimension over by [A4]. The restriction has rank at most , so by [A3] there is with and . Writing with by [A4], the expansion of [A1] and orthonormality of the give , the inequality because for by the nonincreasing order of the singular values. Hence . As was arbitrary among the finite-rank operators with , [A5] gives .
Conclusion. Steps 1.1 and 1.2 give for every ; in the finite-rank case with both sides are by [A1] and [step 1.1], and for the equality reads .
Compact operator iff approximation numbers tend to zero
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be real or complex Hilbert spaces and let be a bounded linear operator (A bounded linear operator between normed spaces, Hilbert space). Put , and for put (Greatest lower bound (infimum), The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis), where only finite-rank are admitted. The error set contains by taking and is bounded below by , so its real infimum exists by Every nonempty set bounded below has an infimum. Thus is a sequence in the library's zero-based convention. Then is compact (Compact linear operator) if and only if (Convergence of a sequence in a metric space: iff in ). If is compact, then for every , where is the zero-padded singular-value sequence of (Absolute value and singular values of a compact operator) for every .
Facts & Assumptions
Given: Countable Choice, Hilbert spaces , a bounded and the approximation numbers .
Compact case. If is compact then for all , and : the sequence is nonincreasing with nonnegative terms, is eventually in the finite-rank case, and in the infinite-rank case consists of the positive eigenvalues of listed with multiplicity, which by the spectral theorem have only as accumulation point, so the nonincreasing listing tends to (Singular values equal approximation numbers, Absolute value and singular values of a compact operator, Singular value decomposition for compact operators).
Finite-rank operators are compact. A bounded finite-rank operator is compact; a norm limit of compact operators with Banach target is compact under ; a Hilbert space is a Banach space (Bounded finite rank operators are compact, Norm limit of compact operators is compact, Hilbert space, Banach space).
Infimum and convergence. Every nonempty bounded-below set of reals has a real infimum (Every nonempty set bounded below has an infimum). Each defining error set is nonempty because it contains the error of , and is bounded below by . For every real it has an element : otherwise would be a larger lower bound, contradicting the greatest-lower-bound definition; a sequence of real numbers tends to when for every eventually (Greatest lower bound (infimum), Convergence of a sequence in a metric space: iff in ).
Countable Choice selects one finite-rank approximant for each by applying it to the shifted family indexed by ; assigning then gives a zero-based sequence (The Axiom of Countable Choice ()).
Proof
Given: Countable Choice, the Hilbert spaces , the bounded operator , and the numbers .
Compact implies vanishing. If is compact then [A1] gives for all and along the positive-indexed tail, so the zero-based sequence tends to ; its single value does not affect convergence.
Vanishing implies compact. Assume . Since the infimum defining is over a nonempty set, for each there is with and , by [A3]; countable choice [A4] selects these operators, and we put to obtain a sequence indexed by . Each has finite rank, hence is compact, and because the tail satisfies ; as the target is a Banach space, [A2] makes compact.
Conclusion. Steps 1.1 and 1.2 give the equivalence; the identification in the compact case for every is [A1].
Finite rank operators are norm dense in compact Hilbert space operators
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be real or complex Hilbert spaces and let be a compact operator (Compact linear operator). Relabel the singular system of by positive integers, so its -th vectors correspond to the numerical singular value , for in rank and for every in infinite rank. Let be the zero-padded singular-value sequence (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator). For put with the empty sum . This is a finite-rank bounded operator, and so and is the operator-norm limit of the finite-rank operators . In particular the set of finite-rank operators is norm dense in the set of compact operators : every compact operator is the norm limit of finite-rank operators.
Facts & Assumptions
Given: Countable Choice, a compact , its singular system and the truncations .
SVD data and relabelling. The SVD supplies orthonormal singular systems indexed by the positive singular values with multiplicity, together with the norm-convergent expansion of and the corresponding partial-sum error estimate. In infinite rank its index set is order-isomorphic to the positive integers via ; after this relabelling, and without any choice, the -th coefficient is the uniquely ordered numerical singular value . Thus and whenever the -st positive singular value exists. If , then for and for (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator).
Finite rank. Each truncation is a finite sum of rank-one operators and therefore has finite rank, hence is compact and bounded (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis, Bounded finite rank operators are compact, A bounded linear operator between normed spaces).
Norm test. The operator norm is the unit-ball supremum, so for every unit vector and follows from for all unit vectors (The operator norm as the least bound and as the unit-sphere or unit-ball supremum); limits in operator norm are metric limits (Convergence of a sequence in a metric space: iff in ).
Countable Choice is the standing hypothesis of this pair's Hilbert-space interface (The Axiom of Countable Choice ()).
Proof
Given: Countable Choice, the compact and its finite-rank truncations .
Upper bound. For every : if the -st positive singular value exists then [A1] gives ; otherwise and , so [A1] gives and . This includes , when for every . Hence for every .
Lower bound. Let . If the -st positive singular value exists, then is a unit vector and the expansion [A1] gives , whence by [A3]; otherwise by [A1] and . In every case .
Conclusion. Steps 1.1 and 1.2 give for every , and because is nonincreasing and nonnegative, is eventually in finite rank, and in infinite rank lists the positive eigenvalues of with multiplicity with only as an accumulation point [A1]; each has finite rank by [A2], so the zero-based sequence converges to in operator norm, proving the asserted density statement.
Hilbert Schmidt operators form a two sided ideal
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let , and be real or complex Hilbert spaces and let be a Hilbert basis of (supplied as data), with the Hilbert–Schmidt square-sum and the Hilbert–Schmidt norm of Hilbert–Schmidt operator and Hilbert–Schmidt norm. Then:
- if are Hilbert–Schmidt relative to , then so is for all scalars , and
- if is Hilbert–Schmidt relative to , then for every Hilbert basis of the adjoint is Hilbert–Schmidt relative to and
- if is Hilbert–Schmidt relative to and , are bounded with a Hilbert basis of and a supplied Hilbert basis of , then is Hilbert–Schmidt relative to and is Hilbert–Schmidt relative to , with and consequently
Facts & Assumptions
Given: Countable Choice, real or complex Hilbert spaces , a Hilbert basis of , Hilbert–Schmidt operators relative to , and bounded operators , .
For claim 3, Hilbert bases of and of are supplied as additional data; their existence is not inferred from Countable Choice.
Hilbert–Schmidt data. For a Hilbert basis of , is the supremum of the finite subsums, is Hilbert–Schmidt relative to when , and then ; the finite-subset supremum splits as for finite (Hilbert–Schmidt operator and Hilbert–Schmidt norm, Square-summable families on an arbitrary index set and the space ).
Adjoint invariance. For every Hilbert basis of and of one has , and membership and norms agree across all such bases (The Hilbert–Schmidt norm is basis independent, Hilbert space).
Bounds. and for bounded operators, and (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces, Composition satisfies |ST|\le|S|,|T|, Hilbert-adjoint identities).
Minkowski in finite dimension. For finitely many vectors of an inner-product space, ; for finitely many pairs of nonnegative reals the Cauchy–Schwarz inequality gives (Cauchy–Schwarz: , with equality exactly for dependent pairs, Real and complex inner-product spaces and their induced length).
Proof
Given: Countable Choice, the Hilbert spaces and bases above, Hilbert–Schmidt relative to , and bounded .
Vector space. For every finite , [A3] and [A5] give , so the finite subsums for are bounded by the square of the last quantity [A1] and is Hilbert–Schmidt relative to with the asserted norm bound.
Adjoint. For every Hilbert basis of , by [A2], so is Hilbert–Schmidt relative to and .
Left multiplication. For finite , [A3] gives , so is Hilbert–Schmidt relative to with .
Right multiplication. Let and be the supplied Hilbert bases of and , respectively, and put , so by [A3]. By [step 1.2], is Hilbert–Schmidt relative to , and [step 1.3] applied to the left multiplication gives that is Hilbert–Schmidt relative to with Now apply [step 1.2] to the Hilbert–Schmidt operator , using as its domain basis and as its codomain basis. It follows that is Hilbert–Schmidt relative to and has Hilbert–Schmidt norm . Since by [A3], is Hilbert–Schmidt relative to and
Both-sided bound. Combining [step 1.3] with in place of and [step 2.1], is Hilbert–Schmidt relative to with .
Conclusion. Claim 1 is [step 1.1], claim 2 is [step 1.2] and claim 3 is the combination of [step 1.3], [step 2.1] and [step 3.1]; no Hilbert basis is assumed to exist, since , and are supplied as data and only the finite-subset supremum definition of [A1] and the invariance theorem [A2] are used.
Trace class operator
Definition
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be Hilbert spaces over the same field (Hilbert space) and let be a compact operator (Compact linear operator) with zero-padded singular-value sequence (Absolute value and singular values of a compact operator).
Trace class. The operator is trace class when Thus the series in the zero-based convention of Series and absolute convergence in a normed space is formed from the explicit sequence . In that case its trace norm is and is also written . The set of trace-class operators is written .
Immediate consequences. Since (Absolute value and singular values of a compact operator), the terms are nonnegative and whenever is trace class; the zero operator is trace class with . When has finite rank the sequence is zero-padded, the series is the finite sum over the finitely many positive eigenvalues of counted with multiplicity (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis), and every finite-rank operator is compact (Bounded finite rank operators are compact) and therefore trace class; in particular every operator with finite-dimensional range and every rank-one operator is trace class. If has infinite rank then for every and the series converges in the summable case. By the necessary condition for convergence of a scalar series (If a series converges then its terms tend to ), a trace-class operator with infinite rank has , hence is a norm limit of finite-rank operators (Singular value decomposition for compact operators).
Choice accounting. Trace class is defined through the singular values of Absolute value and singular values of a compact operator, whose construction uses through the countable selection of finite orthonormal bases of the eigenspaces of and the spectral theorem; no Hilbert basis of the ambient space, and no stronger choice, is used here.
Trace class iff product of two Hilbert Schmidt operators
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be real or complex Hilbert spaces (Hilbert space), let be a Hilbert basis of and a Hilbert basis of (both supplied as data), and let be compact (Compact linear operator). Then:
- (factorization of a trace-class operator) if is trace class (Trace class operator), then, with , and the singular system of (Absolute value and singular values of a compact operator, Singular value decomposition for compact operators), the operators satisfy , are both Hilbert–Schmidt relative to (Hilbert–Schmidt operator and Hilbert–Schmidt norm), and so that : the trace norm is attained by this factorization;
- (products of Hilbert–Schmidt operators are trace class) conversely, if there are a real or complex Hilbert space with a supplied Hilbert basis , a Hilbert–Schmidt operator relative to and a Hilbert–Schmidt operator relative to with , then is trace class and in particular is compact and .
The bases , , are supplied data; no existence of a Hilbert basis is asserted or used, and the adjoint-stability of the Hilbert–Schmidt norm across is the imported invariance theorem.
Facts & Assumptions
Given: Countable Choice, Hilbert spaces , supplied Hilbert bases of , of , of , and a compact .
Trace norm. is trace class exactly when , and then and (Trace class operator, Absolute value and singular values of a compact operator).
SVD and the positive square root. With the index set of positive singular values, in norm, , and orthonormal, is the partial isometry with on extended by zero on , , and is isometric on with range ; moreover . Since is compact, self-adjoint and positive, it has a compact self-adjoint positive square root , which acts by on each and by zero on (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator, Positive square root of a compact positive operator).
Hilbert–Schmidt calculus. The Hilbert–Schmidt norm is basis-independent and adjoint-stable, and for bounded ; a Hilbert–Schmidt operator is compact; composites of compact operators with bounded ones are compact (Hilbert Schmidt operators form a two sided ideal, The Hilbert–Schmidt norm is basis independent, Hilbert–Schmidt operators are compact, Hilbert–Schmidt operator and Hilbert–Schmidt norm, Compositions with a compact operator are compact, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
Parseval, Bessel, collapse of suprema. For a Hilbert basis and any vector, the squared norm is the sum of the squared moduli of the coefficients; Bessel's inequality bounds finite coefficient sums for orthonormal families; for nonnegative families indexed by two sets the finite-subset suprema may be interchanged, (Parseval equivalences for an orthonormal family, The finite Bessel inequality and best approximation by a finite orthonormal family, Square-summable families on an arbitrary index set and the space , Orthonormal families, complete orthonormal systems and Hilbert bases).
Cauchy–Schwarz and boundedness. and the pairing is linear in the first argument and conjugate-linear in the second; and (Cauchy–Schwarz: , with equality exactly for dependent pairs, Real and complex inner-product spaces and their induced length, A bounded linear operator between normed spaces, Hilbert-adjoint identities, Convergence of a sequence in a metric space: iff in ).
Proof
Products of Hilbert–Schmidt operators are trace class. Assume with Hilbert–Schmidt relative to and Hilbert–Schmidt relative to . Then is compact [A3], so is compact [A3] and [A2] applies to with singular system ; for every finite , and writing [A5], finite Cauchy–Schwarz in gives , the last inequality because is an orthonormal family in with a Hilbert basis available so by Bessel and Parseval [A4], and likewise for and ; the equality is the adjoint-stability of the Hilbert–Schmidt norm [A3]. Taking the supremum over finite gives , so is trace class with the asserted bound, and is [A1].
The Hilbert–Schmidt norms of the two factors of a trace-class operator. Assume now that is trace class and put , . First, is self-adjoint with and kills and preserves [A2], so and for every . Hence for the fixed Hilbert basis of , using Parseval for each and the interchange of nonnegative suprema [A4], , so is Hilbert–Schmidt relative to with [A1]. Second, by [A2] every lies in , on which is isometric with values in , so for every ; therefore , so is Hilbert–Schmidt relative to with as well. Finally by [A2].
Conclusion. Claim 2 is [step 1.1], claim 1 is [step 1.2]; the factorization of [step 1.2] has and and attains equality because both norms equal .
Nuclear series characterizes trace norm
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let and be Hilbert spaces over the same real or complex scalar field and let be compact (Compact linear operator). Then is trace class (Trace class operator) if and only if there are families in and in , indexed by the positive integers, with such that the zero-based sequence of finite-rank operators defined by and converges to in operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Convergence of a sequence in a metric space: iff in ). Here the displayed scalar series is, under the library convention, the series of the sequence . In that case the infimum being over all such nuclear representations of , with the same zero-based shift understood in every displayed sum, and the infimum is attained: using its positive-integer index set and padding finite rank by zeros, the singular-value series is a nuclear representation with sum .
Facts & Assumptions
Given: Countable Choice, real or complex Hilbert spaces over the same field, and compact . Nuclear data are assumed only in the reverse implication.
The SVD has in infinite rank and in rank , including when . It supplies orthonormal , , and operator-norm convergence of to (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator).
Trace class and its norm are defined by the sum of the zero-padded positive-indexed singular values, equivalently by the zero-indexed sequence (Trace class operator).
The operator norm bounds , and norm convergence means these norms of differences tend to zero (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces, Convergence of a sequence in a metric space: iff in ).
The pairing is linear in the first argument and conjugate-linear in the second. Cauchy–Schwarz gives , and finite Bessel sums are bounded by the squared norm (Real and complex inner-product spaces and their induced length, Cauchy–Schwarz: , with equality exactly for dependent pairs, The finite Bessel inequality and best approximation by a finite orthonormal family).
An infimum is a greatest lower bound; a member of a set that is also a lower bound is consequently its infimum (Greatest lower bound (infimum)).
A nondecreasing real sequence bounded above converges to the supremum of its range (A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum).
Proof
Suppose is trace class. For every set and . Because is real, conjugate-linearity gives . In finite rank put for ; in rank zero use zero families throughout. In infinite rank already consists of all positive integers, so no index shift and no vector is used. Orthonormality gives for . The shifted zero-based sum is , and the zero-based partial-sum sequence with converges in operator norm by [A1].
Conversely assume given positive-integer-indexed families with in the shifted sense just specified, and let the zero-based sequence converge to in operator norm. Each term is linear and bounded by using [A5], so is bounded and its range lies in the finite span of (the zero subspace when ). The given compactness of licenses [A1]; no new compactness theorem is needed.
Fix a finite . Since , put . For every , finite rearrangement (with the sum empty at ) gives Finite Cauchy–Schwarz, applied to the vectors of absolute values in , and Bessel give Thus . Also by [A3] and [A5], so the zero-based sequence converges to and . For empty this says ; otherwise a hypothetical contradicts the displayed error bound for sufficiently large .
Take for each . The zero-padded singular-value partial sums are nondecreasing, start at zero, and are bounded above by by step 2.1. Therefore [A7] makes their series converge to a value at most . By [A2], is trace class and .
Steps 1.1 and 3.1 prove the equivalence. For trace-class , the set of nuclear-representation sums is nonempty by step 1.1, every such sum is at least by step 3.1, and step 1.1 attains this bound. It is therefore the infimum by [A6]. All sequences used in the reverse implication were given; the forward implication spends only the Countable Choice already assumed by the SVD.
Trace class is a two sided Banach operator ideal
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let , and be real or complex Hilbert spaces (Hilbert space). Then:
- the trace-class operators (Trace class operator) form a linear subspace of on which is a norm, and (The operator norm as the least bound and as the unit-sphere or unit-ball supremum);
- if and , are bounded linear operators on Hilbert spaces , then and
- is a Banach space: every -Cauchy sequence in has a limit in to which it converges in (Banach space, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, Metric space: iff , symmetry, and the triangle inequality; pseudometric and ultrametric).
Facts & Assumptions
Given: Countable Choice, Hilbert spaces , a trace-class operator , bounded operators , and the ideal and nuclear-series results.
Nuclear characterization. For a compact operator , trace class is equivalent to having a nuclear representation (operator-norm convergence, ); the trace norm is the infimum of the nuclear sums and is attained by the singular series. In particular, for trace-class , is zero-padded and is bounded by , because (Nuclear series characterizes trace norm, Trace class operator, Absolute value and singular values of a compact operator).
Infimum and series. The infimum of a nonempty bounded-below set of reals is its greatest lower bound, so for every there is an element below (Greatest lower bound (infimum)). Convergence of the zero-based partial-sum sequences occurring below is interpreted as in Convergence of a sequence in a metric space: iff in .
Operator and adjoint calculus. For composable bounded operators, ; a bounded operator between Hilbert spaces has a bounded adjoint with (Composition satisfies |ST|\le|S|,|T|, Hilbert-adjoint identities, A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
Cauchy sequences and subsequences. A sequence in a metric space is Cauchy when for every real there is with for ; under one may choose indices with , and a Cauchy sequence with a convergent subsequence converges to the same limit (Metric space: iff , symmetry, and the triangle inequality; pseudometric and ultrametric, Convergence of a sequence in a metric space: iff in , The Axiom of Countable Choice ()).
Compactness of nuclear limits. Finite-rank bounded operators are compact, and under an operator-norm limit of compact operators into a Banach space is compact (Bounded finite rank operators are compact, Norm limit of compact operators is compact, Hilbert space).
Proof
Given: Countable Choice, the Hilbert spaces, the trace-class and bounded .
Operator norm dominated by the trace norm. For trace-class , [A1] gives .
Vector-space structure and triangle inequality. Let be trace class and scalars. Given , [A1] and [A2] provide nuclear representations of and with sums and ; interleaving them, after multiplying the first by and the second by , gives a nuclear series converging in operator norm to with sum . Its partial sums have finite rank, so [A5] makes compact; [A1] now makes it trace class and bounds its trace norm by that nuclear sum. Letting gives ; homogeneity follows by also applying the bound to when (and is immediate for ), and the triangle inequality is the case . The norm is definite: if then by [A1], so ; it is nonnegative by definition.
Two-sided ideal estimate. Let be nuclear and let be bounded. Then for every , , and the finite-rank partial sums converge to in operator norm because composition is operator-norm continuous [A3]. Their nuclear sum satisfies by [A3]. Thus is compact by [A5], and [A1] makes it trace class with trace norm bounded by this sum; taking the infimum over nuclear representations of gives .
Completeness. Let be -Cauchy. Choose a subsequence with [A4] and write . For and each choose by [A2] nuclear representations with sums at most and respectively. Flattening these countably many positive-integer-indexed series by a fixed pairing of positive integers produces one nuclear series with total sum at most . The nuclear-tail estimate makes its finite-rank partial sums converge in operator norm to a bounded operator , and [A5] makes compact; [A1] therefore makes trace class. Absolute operator-norm convergence permits regrouping, and the grouped partial sums are , so in operator norm. For every the tail representation made from gives by [A1] and [step 1.2]; hence in , and by [A4] the original Cauchy sequence converges to in .
Conclusion. Claim 1 is [step 1.1] and [step 1.2], claim 2 is [step 1.3], and claim 3 is [step 2.1]; together with is a normed space complete in its norm, that is a Banach space.
Trace of a trace class operator
Definition
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space), let be a trace-class operator on (Trace class operator) and let be a Hilbert basis of , supplied as data (Orthonormal families, complete orthonormal systems and Hilbert bases; existence of such a basis is not asserted here). The trace of relative to is the sum of the scalar family in the finite-subset-net sense of Square-summable families on an arbitrary index set and the space .
The family is absolutely summable, uniformly in . Let be the singular-value series (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator) with , orthonormal and (Trace class operator). For every the series converges absolutely, because the moduli are bounded by by Cauchy–Schwarz and Bessel (Cauchy–Schwarz: , with equality exactly for dependent pairs, The finite Bessel inequality and best approximation by a finite orthonormal family). Consequently, for every finite , using the interchange of finite sums with the nonnegative finite-subset supremum over and then finite Cauchy–Schwarz and Bessel twice, because for each the two finite coefficient sums are bounded by and (finite Bessel). Hence the finite subsums of are bounded by , the family is absolutely summable, and an absolutely summable family of scalars is summable in ZF with (Square-summable families on an arbitrary index set and the space ). This justifies the notation before any basis-independence statement.
Choice accounting and status. Only a supplied basis and the singular values of are used; no Hilbert basis of is assumed to exist, and the next theorem proves that does not depend on the supplied basis and equals the basis-free trace of (Trace is absolutely convergent and basis independent).
Trace is absolutely convergent and basis independent
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space) and let be trace class (Trace class operator). Then:
- for every nuclear representation of (operator-norm convergence of the partial sums, ), the scalar series converges absolutely and
- the sum depends only on ; denoting it , one has for every nuclear representation of . This defines without assuming that has a Hilbert basis;
- for every supplied Hilbert basis of , (Trace of a trace class operator);
- the trace is linear in the trace-class variable and bounded by the trace norm: for trace-class and scalars , and .
Facts & Assumptions
Given: Countable Choice, a Hilbert space , a trace-class , its nuclear representations, and the supplied bases.
Nuclear representations exist and compute the trace norm. trace class means that has a nuclear representation; the SVD series is one, and is the infimum of the nuclear sums (Nuclear series characterizes trace norm, Trace class operator, Singular value decomposition for compact operators, Absolute value and singular values of a compact operator).
Absolute convergence tools. A nonnegative family has a finite-sum supremum; if the finite subsums are bounded by then the family is summable with sum at most , and sums of finite subfamilies of a nonnegative family are bounded by the full sum; for a scalar family, absolute summability implies summability with bound. Suprema over finite subsets of two index sets commute. (Square-summable families on an arbitrary index set and the space , Convergence of a sequence in a metric space: iff in )
Parseval, Bessel, separable bases. For a Hilbert basis of a closed subspace and , with ; for an orthonormal family and any vector the finite coefficient sums obey Bessel; a closed subspace of with a given countable dense sequence has a finite or countable Hilbert basis obtained from that sequence by Gram–Schmidt, with no choice (Parseval equivalences for an orthonormal family, Fourier expansion in a Hilbert space, The finite Bessel inequality and best approximation by a finite orthonormal family, A Hilbert space with a dense sequence has a finite or countable orthonormal basis, Orthonormal families, complete orthonormal systems and Hilbert bases, Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets, Finite, countably infinite, countable, uncountable).
Cauchy–Schwarz and pairing. , the pairing is linear in the first argument and conjugate-linear in the second, and (Cauchy–Schwarz: , with equality exactly for dependent pairs, Real and complex inner-product spaces and their induced length, A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).
Countable Choice is the standing hypothesis; the deterministic construction below uses no choice beyond it (The Axiom of Countable Choice ()).
Proof
Given: Countable Choice, the trace-class and nuclear representations , .
The candidate scalar is absolutely convergent. For a nuclear representation, by [A4], so the scalar series converges absolutely with by [A2].
Comparison of two representations. Put , listing the finitely many zero families as the constant zero sequence when necessary. Let when is real and when is complex. Then is a closed subspace with the at most countable dense set of all finite -linear combinations of the listed vectors (which exists without choice), so by [A3] it has a finite or countable Hilbert basis obtained from that sequence by Gram–Schmidt. Both representations show (the value at is a norm limit of combinations of the , respectively ) and for (all coefficients , vanish). For the representation , expanding both factors in the basis by Parseval [A3] and using absolute convergence and the interchange of nonnegative finite-subset suprema [A2], , the inner identity because in norm. The same computation applies to , so both representations have the same scalar sum; since a trace-class operator has at least one nuclear representation by [A1], the scalar is well defined and claim 2 holds.
Agreement with every supplied basis. Let be a Hilbert basis of and let be any nuclear representation of . For each , Parseval in the full space gives and ; hence, by Cauchy–Schwarz for the -sum, interchange of the nonnegative suprema [A2] and the definition of the representation, . Therefore the double sum converges absolutely, its value may be computed in either order, and , the last equality by Parseval applied to the pair in . This is exactly , and it also reproves the absolute summability of required by the definition.
Linearity and the bound. For trace-class with nuclear representations and , the concatenation of scaled by and scaled by is a nuclear representation of with scalar sum by absolute convergence, so ; and for every nuclear representation by [step 1.1], so the infimum characterization [A1] gives .
Conclusion. Claims 1 and 2 are [step 1.1] and [step 1.2], claim 3 is [step 1.3] and claim 4 is [step 2.1]; the definition of uses only nuclear representations of , so no Hilbert basis of is assumed to exist, while claim 3 handles every basis that is supplied.
Cyclicity of the trace
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space), let be trace class (Trace class operator) and let be bounded. Then and are trace class and If are Hilbert–Schmidt relative to a supplied Hilbert basis of (Hilbert–Schmidt operator and Hilbert–Schmidt norm), then the products and are trace class and
Facts & Assumptions
Given: Countable Choice, a Hilbert space , a trace-class , a bounded , and Hilbert–Schmidt relative to a supplied basis .
Trace and its properties. Every trace-class has a nuclear representation and a well-defined trace. For every supplied Hilbert basis , that trace equals the absolutely convergent diagonal sum . It is equal to for every nuclear representation , with and linearity in the trace-class variable; (Trace is absolutely convergent and basis independent, Trace class operator, Nuclear series characterizes trace norm, Trace of a trace class operator).
Trace ideal. Bounded one-sided multiplication preserves trace class, and (Trace class is a two sided Banach operator ideal).
Hilbert–Schmidt products. An operator which is Hilbert–Schmidt relative to a supplied basis is compact. Thus and are compact, and their bounded composites and are compact; the product clause of the factorization theorem then makes both products trace class and gives (Hilbert–Schmidt operators are compact, Compositions with a compact operator are compact, Trace class iff product of two Hilbert Schmidt operators, Hilbert Schmidt operators form a two sided ideal).
SVD and adjoints. The SVD uses only positive singular-value indices, with numerical zero padding beyond finite rank, and its partial sums converge in operator norm. The adjoint is bounded with , and the identity holds (Singular value decomposition for compact operators, Absolute value and singular values of a compact operator, Hilbert-adjoint identities, The Hilbert-space adjoint of a bounded operator).
Parseval and absolute convergence. Parseval's identity holds for supplied Hilbert bases; the finite-subset suprema of nonnegative families may be interchanged; absolutely summable scalar families are summable; and finite Cauchy–Schwarz bounds coefficient sums (Parseval equivalences for an orthonormal family, The finite Bessel inequality and best approximation by a finite orthonormal family, Square-summable families on an arbitrary index set and the space , Cauchy–Schwarz: , with equality exactly for dependent pairs, Real and complex inner-product spaces and their induced length, Orthonormal families, complete orthonormal systems and Hilbert bases, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces, Convergence of a sequence in a metric space: iff in ).
Proof
Given: Countable Choice, the trace-class , bounded , Hilbert–Schmidt relative to .
Products are trace class. and are trace class by [A2] with and .
Cyclicity for rank-one operators. Let for fixed , so that is trace class; then and are nuclear representations with one term, so by [A1] and , and these are equal by the adjoint identity of [A4].
The Hilbert–Schmidt case. Write for , so that basis elements and indices are unambiguous. Let be Hilbert–Schmidt relative to the supplied basis ; by [A3] the products and are trace class. Since is continuous and is a Hilbert basis, for every , and the double family with , has finite total, , by two applications of finite Cauchy–Schwarz, Bessel and Parseval [A5]; For detail, on each finite rectangle , finite Cauchy–Schwarz bounds the absolute sum by , using Bessel in the inner sums. Every finite set of pairs lies in a finite rectangle. The full absolute sum is therefore finite; outside a finite rectangle its tail is arbitrarily small by [A5], which proves that both iterated scalar sums have the same value as the double-family sum. Invoking the diagonal trace formula in [A1] for the trace-class products, , the last equality by the symmetric computation for .
Cyclicity for general trace-class operators. By [A1] take a nuclear representation with and partial sums for , so and in operator norm. Then and : both errors are at most by the operator-norm bound. Their terms give nuclear representations and . Their sums of norm products are at most and , respectively. Both products are already trace class by step 1.1, so [A1] gives by [A4], term by term in absolutely convergent series. No ambient Hilbert basis is used in this part.
Conclusion. The general cyclicity statement is [step 2.1] with [step 1.1], and the Hilbert–Schmidt statement is [step 1.3].
Trace of a positive operator is the sum of its eigenvalues
Statement
Assume the Axiom of Countable Choice (The Axiom of Countable Choice ()). Let be a real or complex Hilbert space (Hilbert space) and let be a self-adjoint positive trace-class operator (Self-adjoint, positive, unitary and normal operators, Trace class operator), so that for every . Let be the positive eigenvalues of listed with multiplicity (Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism) and let be the zero-padded singular-value sequence (Absolute value and singular values of a compact operator). Then for every , and where the sum is over the positive eigenvalues with multiplicity and the equalities are also valid in the finite-rank case (then for beyond the rank). This is the positive compact self-adjoint case only; it is not Lidskii's theorem for arbitrary trace-class operators, which is not claimed here.
Facts & Assumptions
Given: Countable Choice, the Hilbert space , a self-adjoint positive trace-class , its positive eigenvalues with multiplicity and eigenspaces .
Spectral theorem. is compact self-adjoint; its positive eigenvalues have finite-dimensional eigenspaces, the closed linear span of those eigenspaces is , , and in norm, where is the orthogonal projection onto (Spectral theorem for compact self adjoint operators, Eigenspaces of a self adjoint operator are orthogonal, Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism).
Positivity forces nonnegative eigenvalues. If with then , so ; and is self-adjoint with (Self-adjoint, positive, unitary and normal operators, Hilbert-adjoint identities, The Hilbert-space adjoint of a bounded operator, Real and complex inner-product spaces and their induced length).
Absolute value of a positive operator. For compact self-adjoint positive one has , by uniqueness of the compact positive square root applied to the compact self-adjoint positive operator satisfying (Absolute value and singular values of a compact operator, Positive square root of a compact positive operator).
SVD and trace. The singular values of are the positive eigenvalues of with multiplicity; the SVD gives orthonormal in with and ; after zero-padding a finite SVD to positive-integer-indexed coefficient families, the trace of a trace-class operator is for every nuclear representation , and (Absolute value and singular values of a compact operator, Singular value decomposition for compact operators, Trace is absolutely convergent and basis independent, Trace class operator, Nuclear series characterizes trace norm).
Orthonormal bases of eigenspaces. Every finite-dimensional eigenspace has an orthonormal basis, and orthonormal families consist of unit pairwise orthogonal vectors (Every finite-dimensional real or complex inner product space has an orthonormal basis, Orthonormal families, complete orthonormal systems and Hilbert bases, Convergence of a sequence in a metric space: iff in ).
Proof
Given: Countable Choice, the self-adjoint positive trace-class , its positive eigenvalues with multiplicity and their eigenspaces.
. By [A2] is self-adjoint with , so the compact self-adjoint positive operator satisfies ; since the positive square root of a compact self-adjoint positive operator is unique by [A3], and because , the definition yields .
The singular values are the positive eigenvalues. By [A1] and [A2] the eigenvalues of are nonnegative, its positive eigenvalues are exactly the nonzero eigenvalues, and listing them with multiplicity as matches the nonincreasing listing of the positive eigenvalues of with multiplicity required by [step 1.1]; hence for every , with zeros appended once the positive eigenvalues are exhausted (the finite-rank case), and .
The trace equals the eigenvalue sum. For each positive eigenvalue choose an orthonormal basis of by [A5]; the union over the positive eigenvalues is an orthonormal family whose closed span is by [A1] and [A5]. Since by [step 1.1], the SVD of has , and . For put and ; in the finite-rank case , extend these to all positive integers by for (and use the all-zero families when ). With and , the SVD gives in operator norm and . Thus these positive-integer-indexed families are a nuclear representation in the precise sense of [A4], and its trace formula gives .
Conclusion. Steps 2.1 and 3.1 give and ; the computation never chooses a basis of , only orthonormal bases of the finite-dimensional positive eigenspaces, and the identity is stated for self-adjoint positive operators only, as the statement records.
Separable trace-class determinant theorem recorded externally
Statement
Assume the Axiom of Choice (The Axiom of Choice). Let be a separable complex Hilbert space, including the zero space, and let (Trace class operator). For every nonzero eigenvalue of , its algebraic multiplicity is
where the increasing generalized kernels stabilize and this dimension is finite. List all nonzero eigenvalues with those multiplicities; the list is finite or countable and may be empty. With the singular values (Absolute value and singular values of a compact operator), the following results are recorded externally.
- The eigenvalues are absolutely summable and
- There is an entire function such that, locally uniformly in , The empty product is and the empty eigenvalue sum is .
- If finite-rank satisfy , then locally uniformly. For finite-rank , this is the ordinary determinant of for any finite-dimensional subspace containing ; it is independent of .
- One has For every there is with .
- For , and .
- The value vanishes exactly when is not boundedly invertible. If is an eigenvalue, then is a zero of order equal to its algebraic multiplicity.
These assertions include the zero-space conventions: its unique operator has determinant identically , trace , and an empty eigenvalue list.
Remarks
This is a source-backed external theorem, not a local exterior-power or Hadamard-factorization proof. In the cited proof of the zero criterion, the complementary factor is ; a printed omission of in one sentence is not copied here.
Fredholm determinant of a trace-class operator
Definition
proof uses external results not yet established in this library
Assume the Axiom of Choice (The Axiom of Choice). Let be a complex Hilbert space and let (Trace class operator). Choose a nuclear representation and put and . The Fredholm determinant of is
where is the separable determinant recorded in Separable trace-class determinant theorem recorded externally ‡. If , this definition gives .
The well-definedness argument below proves that this restriction preserves the trace, trace norm, nonzero singular values, and nonzero generalized-eigenvalue data, and that the resulting determinant is independent of the nuclear representation and separable reducing support.
Well-definedness
Nuclear representations exist by Nuclear series characterizes trace norm. Finite rational-complex linear combinations of the vectors form a countable dense subset of , so is separable. If , every coefficient in the nuclear series vanishes and ; if , every partial sum and hence lies in the closed space . Thus, using Orthogonal decomposition by a closed subspace,
The restriction is bounded and compact. Indeed, a bounded sequence in is bounded in . Since is compact, its images have a norm-convergent subsequence by Sequential characterization of compact operators; the limit lies in the closed space . The converse direction of that same characterization makes compact. Full AC supplies its DC hypothesis. The same nuclear series, now regarded inside , therefore makes trace class by Nuclear series characterizes trace norm. The nuclear trace formula in Trace is absolutely convergent and basis independent gives .
The block identity gives . Hence is a compact positive square root of , and uniqueness in Positive square root of a compact positive operator gives . Therefore and have the same nonzero singular values, with multiplicities, and .
For every and ,
Consequently all generalized -eigenvectors lie in , and and have identical nonzero eigenvalues, generalized kernels, stabilization indices, and algebraic multiplicities. The external product formula therefore makes independent of the chosen nuclear representation and of every separable closed reducing support on whose orthogonal complement is zero. No arbitrary invariant subspace is asserted to reduce .
Fredholm determinant properties for trace-class operators
Statement
proof uses external results not yet established in this library
Assume the Axiom of Choice. Let be a complex Hilbert space and . The function is entire and, locally uniformly in ,
where all nonzero eigenvalues are listed with their finite algebraic multiplicities, and . It satisfies
and for every there is such that . For trace-class , and
Moreover exactly when is not boundedly invertible, and the zero at has the algebraic multiplicity of .
If finite-rank converge to in trace norm, their ordinary finite-dimensional determinants converge to locally uniformly. Where is invertible,
All assertions include , finite eigenvalue lists and the empty list.
Facts & Assumptions
Given: The Axiom of Choice, a complex Hilbert space , and the displayed trace-class operators.
The arbitrary-space determinant is well defined through a separable reducing support and preserves trace, trace norm, nonzero singular values, and nonzero generalized-eigenvalue data (Fredholm determinant of a trace-class operator).
The separable determinant has the absolute eigenvalue bound , spectral product, growth, continuity, multiplicativity, derivative-at-zero and zero-multiplicity properties recorded externally (Separable trace-class determinant theorem recorded externally ‡).
Trace-class operators form a two-sided ideal (Trace class is a two sided Banach operator ideal).
The trace is basis-independent and agrees with every nuclear trace sum (Trace is absolutely convergent and basis independent).
Proof
Choose the separable reducing support from [F1]. Its restriction has the same trace, trace norm, nonzero singular values and algebraic eigenvalue data as . Every single-operator assertion in the first paragraph, including the zero criterion and zero order, therefore transfers term by term from [F2]. The block identity also proves the equivalence of bounded invertibility.
For trace-class , take one separable closed span of nuclear vectors for both. It reduces , , and all three operators vanish on its orthogonal complement; [F3] supplies the trace-class hypotheses. Apply the external multiplicativity and continuity formulas on this common support and then [F1] to obtain the displayed arbitrary-space formulas.
If finite-rank in trace norm, full AC chooses nuclear representations for the countable family. The closed span of all their input and output vectors and those for is a common separable reducing support. The block argument in [F1] preserves the trace norm of every difference , so the locally uniform finite-rank limit in [F2] applies. For a finite-rank , any finite-dimensional is invariant under , and enlargement adds an identity diagonal block; hence the ordinary determinant is independent of .
Fix with invertible and put , which is trace class by [F3]. Since , step 1.2 gives . Steps 1.1 and [F2] give . Dividing by and taking the limit gives . The operator commutes with and its inverse, so . The zero-space and empty-list conventions follow from [F1] and [F2].
Lidskii trace formula for trace-class operators
Statement
proof uses external results not yet established in this library
Assume the Axiom of Choice. Let be any complex Hilbert space, including , and let . List all nonzero eigenvalues with their finite algebraic multiplicities, where the multiplicity of is the dimension of the stabilized generalized kernel . Then The list is finite or countable and may be empty. No normality, self-adjointness, positivity, or separability of is assumed.
Facts & Assumptions
Given: The Axiom of Choice, a complex Hilbert space , and a trace-class operator .
The determinant definition preserves the nonzero generalized-eigenvalue data under separable-support reduction (Fredholm determinant of a trace-class operator).
The determinant properties give absolute eigenvalue summability, locally uniformly, and (Fredholm determinant properties for trace-class operators).
Proof
By [F1] and [F2], the eigenvalue list has the stated algebraic multiplicities and . For a finite initial product , expansion and the ordered-tuple bound for elementary symmetric sums give . Indeed, every unordered product of distinct absolute eigenvalues occurs times among the ordered -tuples contributing to .
Let . The local product convergence and absolute convergence of from [F2] preserve the bound in step 1.1, so . The left side is by [F2]. The same argument applies to finite and empty lists, with the empty sum equal to and the empty product equal to .
5 · Examples, counterexamples and false statements
None yet.
Sources
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §1.3, Problems 1.19–1.20 (quadratic-form bound and polarization)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §2, Proposition 2.2
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.2, Theorem 3.6 (printed pp. 73–74)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §2, Theorem 2.3
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.2, self-adjointness and orthogonality of eigenspaces
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §2
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.2, reduction of the spectral problem to an invariant complement
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §2, orthogonality and invariant complements
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.2, Theorem 3.7 and Corollaries 3.8–3.9 (printed pp. 74–78)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.2, Corollary 3.9 (printed p. 78)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.2 (support decomposition) and §3.5 (absolute value of a compact operator)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §§2 and 5
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.5, singular values of a compact operator (printed pp. 89–93)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §5
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.5, Theorem 3.17 (printed pp. 90–92)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §§4–5
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.5, Lemma 3.19 (printed pp. 92–93)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.5, Lemma 3.19 and its converse (printed pp. 92–93)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.5, finite-rank singular truncations (printed pp. 90–93)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, Lemmas 3.23–3.25 (printed pp. 93–97)
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- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §5, Propositions 2.8–2.9
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- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, Lemma 3.27 (printed pp. 97–98)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, Lemma 3.28 (printed pp. 98–99)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §5, cyclicity exercises
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, positive trace and the eigenvalue sum (printed pp. 97–100)
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