Alphabeta MathFredholm Determinants and the Lidskii Trace Formula: Examples
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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
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Combinatorics
Complex Analysis
- Analyticity of Holomorphic Functions; Liouville and Morera
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Functional Analysis
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Compact Self Adjoint Hilbert Schmidt and Trace Class Operators
- Complex Norms and Realification
- Complex Subspace Realification
- Dual Spaces Adjoint Operators and Annihilators
- Finite Dimensional Normed Spaces and Riesz Lemma
- Fredholm Determinants and the Lidskii Trace Formula
- Geometric Hahn Banach and Convex Separation
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Orthonormal Bases, Parseval and Fourier Series
- Square-Integrable Kernels and Hilbert–Schmidt Compactness
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Weak and Weak Star Topologies
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
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- Matrices, the Matrix of a Linear Map, and Change of Basis
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- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
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- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Bounded Variation and the Riemann–Stieltjes Integral
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Counterexample
An invariant subspace need not reduce an operator
Example
The square of the Volterra operator has zero trace
Example
Diagonal trace-class Fredholm determinant
Example
Fredholm determinant of a finite-rank operator
Sources
- Kostenko, Trace Ideals with Applications, §3.4
- van Neerven, Functional Analysis, §14.5.a
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, App. B §§B.5–B.6
- Gerald Teschl, Topics in Real and Functional Analysis, §6.1, Volterra operator example, equations (6.23)–(6.26), printed pp. 167–168; comparison only (the example acts on C([0,1]) and leaves the power estimate as Problem 6.7)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.5–§3.6, diagonal operators and Schatten classes
- Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, Appendix B §§B.5–B.6