TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Hilbert-adjoint identities

Statement

Assume the Axiom of Countable Choice. Let $H, K, L$ be real or complex Hilbert spaces and let $R, S \in B (H, K)$ and $T \in B (K, L)$ be bounded linear operators. Then the Hilbert adjoints satisfy:

$(a R + b S)^{*} = \overline{a} R^{*} + \overline{b} S^{*}$ for scalars $a, b$ , and the adjoint of an operator is unique;
$(T S)^{*} = S^{*} T^{*}$ ;
$T^{* *} = T$ and $∥ T^{*} ∥ = ∥ T ∥$ ;
$∥ T^{*} T ∥ = ∥ T ∥^{2}$ .

Facts & Assumptions

[A1]

The Hilbert adjoint of $T \in B (K, L)$ is the unique map $T^{*} : L \to K$ with $⟨ T x, y ⟩_{L} = ⟨ x, T^{*} y ⟩_{K}$ for all $x, y$ (The Hilbert-space adjoint of a bounded operator).

[A2]

The pairing is linear in the first argument, conjugate-linear in the second, conjugate symmetric, and $⟨ v, v ⟩ = 0$ implies $v = 0$ (Real and complex inner-product spaces and their induced length).

[A3]

The operator norm satisfies $∥ T u ∥ \leq ∥ T ∥ ∥ u ∥$ and is the unit-ball supremum (The operator norm as the least bound and as the unit-sphere or unit-ball supremum), while composition obeys $∥ U V ∥ \leq ∥ U ∥ ∥ V ∥$ (Composition satisfies |ST|\le|S|,|T|); a linear map is bounded if it admits a finite constant $C \geq 0$ with $∥ U y ∥ \leq C ∥ y ∥$ for every $y$ (A bounded linear operator between normed spaces).

[A4]

Cauchy–Schwarz gives $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[A5]

Countable Choice is the hypothesis of the Riesz construction of adjoints (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: Countable Choice, Hilbert spaces $H, K, L$ and bounded operators $R, S \in B (H, K)$ , $T \in B (K, L)$ .

1.1

First let $U : X \to Y$ be any bounded linear operator between Hilbert spaces over the same scalar field. The map $U^{*} : Y \to X$ exists by the adjoint construction. For $y, z \in Y$ , scalars $a, b$ and $x \in X$ , its identity gives $⟨ x, U^{*} (a y + b z) ⟩ = ⟨ U x, a y + b z ⟩ = \overline{a} ⟨ x, U^{*} y ⟩ + \overline{b} ⟨ x, U^{*} z ⟩ = ⟨ x, a U^{*} y + b U^{*} z ⟩$ . Positive definiteness applied to the difference proves that $U^{*}$ is linear. Also $∥ U^{*} y ∥^{2} = ⟨ U (U^{*} y), y ⟩ \leq ∣ ⟨ U (U^{*} y), y ⟩ ∣ \leq ∥ U ∥ ∥ U^{*} y ∥ ∥ y ∥$ , so division when $U^{*} y \neq 0$ , and the trivial inequality otherwise, give $∥ U^{*} y ∥ \leq ∥ U ∥ ∥ y ∥$ . Thus $U^{*}$ is bounded and its now-defined operator norm satisfies $∥ U^{*} ∥ \leq ∥ U ∥$ . This applies to each bounded operator used below, including an adjoint once its boundedness has been established.

A1A2A3A4A5algebra

2.1

For uniqueness of the adjoint and conjugate-linearity in the operator, let $U, V \in B (K, H)$ both satisfy the defining adjoint identity for the same operator in $B (H, K)$ . For $y \in K$ , one has $⟨ x, (U - V) y ⟩_{H} = 0$ for every $x \in H$ ; taking $x = (U - V) y$ gives $(U - V) y = 0$ . For scalars $a, b$ , the identities $⟨ (a R + b S) x, y ⟩_{K} = a ⟨ R x, y ⟩_{K} + b ⟨ S x, y ⟩_{K} = ⟨ x, \overline{a} R^{*} y + \overline{b} S^{*} y ⟩_{H}$ hold for all $x \in H$ and $y \in K$ , so $(a R + b S)^{*} = \overline{a} R^{*} + \overline{b} S^{*}$ .

step 1.1A1A2A5

3.1

Composition and involution: for $x \in H$ and $y \in L$ , $⟨ x, S^{*} T^{*} y ⟩_{H} = ⟨ S x, T^{*} y ⟩_{K} = ⟨ T S x, y ⟩_{L}$ , so $(T S)^{*} = S^{*} T^{*}$ by uniqueness. Likewise, for $x \in K$ and $y \in L$ , the defining identity for $T^{*}$ gives $⟨ T^{*} y, x ⟩_{K} = ⟨ y, T^{* *} x ⟩_{L}$ ; conjugate symmetry and the defining identity for $T$ give $⟨ T^{* *} x, y ⟩_{L} = ⟨ x, T^{*} y ⟩_{K} = ⟨ T x, y ⟩_{L}$ , so $T^{* *} = T$ by uniqueness.

step 1.1step 2.1A1A2

4.1

Norms: Cauchy–Schwarz gives $∥ T^{*} y ∥^{2} = ⟨ T^{*} y, T^{*} y ⟩_{K} = ⟨ T (T^{*} y), y ⟩_{L} \leq ∥ T ∥ ∥ T^{*} y ∥ ∥ y ∥$ , hence $∥ T^{*} y ∥ \leq ∥ T ∥ ∥ y ∥$ (trivially when $T^{*} y = 0$ ) and $∥ T^{*} ∥ \leq ∥ T ∥$ ; applying this to $T^{*}$ and using $T^{* *} = T$ gives $∥ T^{*} ∥ = ∥ T ∥$ . Moreover $∥ T^{*} T ∥ \leq ∥ T^{*} ∥ ∥ T ∥ = ∥ T ∥^{2}$ , while for $∥ x ∥ \leq 1$ one has $∥ T x ∥^{2} = ⟨ x, T^{*} T x ⟩_{K} \leq ∥ x ∥ ∥ T^{*} T x ∥ \leq ∥ T^{*} T ∥$ , so $∥ T ∥^{2} \leq ∥ T^{*} T ∥$ and hence $∥ T^{*} T ∥ = ∥ T ∥^{2}$ .

step 1.1step 3.1A3A4algebra∎

Depends on

Used by

Orthonormal eigenbasis for a compact self adjoint operator Corollary
Spectral projections and resolution of the identity Corollary
A quasinilpotent operator need not be zero Counterexample
Self adjointness cannot be dropped from the order calculus Counterexample
The unilateral shift obstructs a cyclic linear trace extension Counterexample
Absolute value and singular values of a compact operator Definition
Absolute value of a bounded operator Definition
C star algebra generated by a normal operator Definition
Isometry coisometry and partial isometry Definition
Order on bounded self adjoint operators Definition
Projection valued measure Definition
Self-adjoint, positive, unitary and normal operators Definition
Adjoint, norm and trace of an operator of rank at most one Example
Integral operator trace under a valid diagonal hypothesis Example
Polar decomposition of the unilateral shift Example
Square root and absolute value of a matrix Example
Bounded Hilbert operators form a C star algebra Lemma
Continuous functional calculus produces a regular PVM Lemma
Eigenspaces of a self adjoint operator are orthogonal Lemma
Kernel–range orthogonality for Hilbert adjoints Lemma
Maximal orthogonal family of cyclic reducing subspaces Lemma
Norm of a self adjoint operator from its quadratic form Lemma
Norm point of a compact self adjoint operator is an eigenvalue up to sign Lemma
Orthogonal complement of an eigenspace is invariant Lemma
Polynomial calculus is isometric for self adjoint operators Lemma
Positive square root of a compact positive operator Lemma
Spectral convolution eigenspaces are finite-dimensional and invariant Lemma
Spectrum of a positive operator is nonnegative Lemma
Spectrum of a self adjoint operator is real Lemma
Unitary intertwiners preserve direct-integral fiber dimension Lemma
Borel functional calculus for bounded normal operators Theorem
Cayley correspondence between self-adjoint operators and unitaries Theorem
Continuous functional calculus properties Theorem
Cyclicity of the trace Theorem
Hilbert Schmidt operators form a two sided ideal Theorem
L two kernels give Hilbert–Schmidt operators Theorem
Numerical radius is an equivalent operator norm Theorem
Partial isometry characterizations Theorem
Positive square root Theorem
Pvm integral is a star homomorphism Theorem

…and 6 more results.

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.36 and Lemma 5.38, pp.237–238 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 185 (standard reference, not scraped)