Alphabeta MathFredholm Determinants and the Lidskii Trace Formula
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
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Combinatorics
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Linear Algebra
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2 · Logical flowchart
3 · Items
Definition
Algebraic multiplicity of a nonzero compact-operator eigenvalue
Lemma
Finite-rank orthogonal compressions converge in trace norm
Definition
Hilbert exterior powers and induced operators
Lemma
Trace-norm bound for exterior powers of trace-class operators
Lemma
Weyl product and sum inequalities for compact operators
Lemma
Diagonal trace-class operators on ℓ²(ℕ,ℂ)
Lemma
Local separable trace-class determinant construction
Lemma
Trace-norm continuity, growth and multiplicativity of the local determinant
Lemma
Logarithmic derivative of the local Fredholm determinant
Lemma
Zeros of the local Fredholm determinant
Lemma
A quasinilpotent trace-class operator has zero trace
Lemma
Trace decomposition through generalized eigenspaces and the invariant quotient
Lemma
Spectral product from traces of powers
Lemma
Arbitrary-Hilbert Fredholm determinant from a separable reducing support
Definition
Fredholm determinant of a trace-class operator
Proposition
Fredholm determinant properties for trace-class operators
Theorem
Lidskii trace formula for trace-class operators
Sources
- Kostenko, Trace Ideals with Applications, §3.4
- van Neerven, Functional Analysis, §14.5.a
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §§B.5–B.6
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, App. B §§B.5–B.6
- Kostenko, Trace Ideals with Applications, §3.4.1, equations (3.4.7)–(3.4.10) and Lemma 3.4.2, printed pp. 34–36 / PDF pp. 43–44
- van Neerven, Functional Analysis, §14.5.a and Appendix B
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §B.5
- Kostenko, Trace Ideals with Applications, §3.4.2, Theorem 3.4.2 and proof, printed pp. 36–37 (PDF pp. 45–46)
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §B.5.1, Proposition B.23, Lemma B.24, and Proposition B.25, PDF pp. 506–508
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.5–§3.6, diagonal operators and Schatten classes
- Kostenko, Trace Ideals with Applications, §3.4.3, Proposition 3.4.3 and Corollary 3.4.1, printed pp. 38–39 (PDF pp. 47–48)
- van Neerven, Functional Analysis, §14.5.a, Definition 14.34 and Lemma 14.38, printed pp. 584–587 (PDF pp. 596–599)
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §B.5.2, finite-rank determinant reduction, PDF p. 508
- Kostenko, Trace Ideals with Applications, §3.4.3, Corollary 3.4.1, Theorem 3.4.4 and Corollary 3.4.2, printed pp. 38–40 (PDF pp. 47–49)
- van Neerven, Functional Analysis, §14.5.a, Lemmas 14.35–14.39, printed pp. 585–587 (PDF pp. 597–599)
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §§B.5.2–B.5.3, Propositions B.27 and B.29, PDF pp. 509–511
- Kostenko, Trace Ideals with Applications, §3.4.3, Corollary 3.4.2, printed p. 40 (PDF p. 49)
- van Neerven, Functional Analysis, §14.5.a, Definition 14.34 and Lemma 14.39, printed pp. 585 and 587 (PDF pp. 597 and 599)
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §B.5.2, Lemma B.26, PDF p. 509
- Kostenko, Trace Ideals with Applications, §3.4.4, Theorem 3.4.6, printed pp. 40–41 (PDF pp. 49–50)
- Kostenko, Trace Ideals with Applications, §3.4.4, Theorem 3.4.5 and proof of Theorem 3.4.7, printed pp. 40–42 (PDF pp. 49–51); comparison only
- Kostenko, Trace Ideals with Applications, §3.4.4, Theorem 3.4.7 proof, printed pp. 41–42 (PDF pp. 50–51); comparison only
- van Neerven, Functional Analysis, Proposition 14.22, printed pp. 574–575 (PDF pp. 585–586), and Theorem 14.33 proof, §14.5.a, printed pp. 583–591 (PDF pp. 594–602); comparison only
- Kostenko, Trace Ideals with Applications, §3.4.3–3.4.4, Theorem 3.4.7 proof, printed pp. 38–42 (PDF pp. 47–50)
- van Neerven, Functional Analysis, §14.5.a, Theorem 14.33 through Theorem 14.43, printed pp. 583–591 (PDF pp. 595–603)
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §§B.5–B.6, Propositions B.30–B.31
- Aleksey Kostenko, Trace Ideals with Applications — Section 3.4, printed pp. 34–41
- Aleksey Kostenko, Trace Ideals with Applications — Sections 3.4–3.5, printed pp. 34–45
- Aleksey Kostenko, Trace Ideals with Applications — Theorem 3.4.7, printed p. 41