Alphabeta MathCompact Self Adjoint Hilbert Schmidt and Trace Class Operators
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
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- Sylow's Theorems, p-Groups and Nilpotent Groups
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
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Linear Algebra
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Number Theory
Real Analysis
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Set Theory
2 · Logical flowchart
3 · Items
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
Eigenspaces of a self adjoint operator are orthogonal
Lemma
Orthogonal complement of an eigenspace is invariant
Theorem
Spectral theorem for compact self adjoint operators
Corollary
Orthonormal eigenbasis for a compact self adjoint operator
Lemma
Positive square root of a compact positive operator
Definition
Absolute value and singular values of a compact operator
Theorem
Singular value decomposition for compact operators
Lemma
Singular values equal approximation numbers
Corollary
Compact operator iff approximation numbers tend to zero
Corollary
Finite rank operators are norm dense in compact Hilbert space operators
Theorem
Hilbert Schmidt operators form a two sided ideal
Definition
Trace class operator
Theorem
Trace class iff product of two Hilbert Schmidt operators
Lemma
Nuclear series characterizes trace norm
Theorem
Trace class is a two sided Banach operator ideal
Definition
Trace of a trace class operator
Theorem
Trace is absolutely convergent and basis independent
Theorem
Cyclicity of the trace
Theorem
Trace of a positive operator is the sum of its eigenvalues
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)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §5, Proposition 2.8
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, trace class operators (printed pp. 93–96)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, Lemma 3.26 (printed pp. 95–97)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §5, Propositions 2.8–2.9
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, Lemma 3.29 (printed pp. 98–100)
- Anthony W. Knapp, Advanced Real Analysis — Chapter II, §5, Proposition 2.9
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.6, Lemmas 3.26 and 3.28–3.29 (printed pp. 95–100)
- 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)