Alphabeta MathBanach Algebras Spectrum and Holomorphic Functional Calculus
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- 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
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 Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Complex Norms and Realification
- Complex Subspace Realification
- Dual Spaces Adjoint Operators and Annihilators
- Finite Dimensional Normed Spaces and Riesz Lemma
- Geometric Hahn Banach and Convex Separation
- Normed and Banach Spaces
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
Linear Algebra
- Complexification, Realification and Real Structures
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- 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 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
2 · Logical flowchart
3 · Items
Definition
Unital Banach algebra
Definition
Invertible element and general linear group of a Banach algebra
Lemma
Neumann series
Theorem
Invertible group is open and inversion is continuous
Definition
Spectrum and resolvent set in a Banach algebra
Lemma
Resolvent identity
Theorem
Resolvent is Banach-valued holomorphic
Theorem
Spectrum is nonempty compact and norm bounded
Definition
Spectral radius
Theorem
Polynomial spectral mapping
Lemma
Submultiplicative root limit
Theorem
Spectral radius formula
Lemma
Canonical Banach complexification of a real Banach space
Definition
Complexification and spectrum of a real operator
Theorem
Gelfand-Mazur
Definition
Banach algebra valued contour integral
Lemma
Contour integral commutes with bounded linear maps
Lemma
Banach-valued Cauchy integral vanishes
Lemma
Admissible cycle around a compact plane set
Definition
Holomorphic functional calculus
Lemma
Holomorphic functional calculus is contour independent
Theorem
Holomorphic functional calculus homomorphism
Theorem
Holomorphic spectral mapping and composition
Definition
Riesz spectral projection
Theorem
Riesz spectral projection properties
Definition
Calkin algebra
Corollary
Atkinson in Calkin algebra language
Definition
Point continuous and residual spectrum
Definition
Approximate point and compression spectrum
Lemma
Relations among the five spectral parts
Theorem
Boundary of spectrum lies in approximate point spectrum
Sources
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.1.1 (Banach algebras), printed pp. 209–214
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Chapter 2 §2.1, printed pp. 19–24
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.1.1 (invertible elements and the Neumann series), printed pp. 209–214
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Corollary 1.51 and §5.1.1, printed pp. 34 and 209–214
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 1.49 and §5.1.1, printed pp. 33 and 209–214
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.2.1, printed pp. 219–222
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Definition 2.3.1 and §2.3, printed pp. 30–33
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Lemma 5.19, printed p. 221
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Chapter 2 §2.3, printed pp. 30–33
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Lemma 5.19, printed pp. 221–222
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 5.20, printed pp. 222–223
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — §2.3, printed pp. 30–33
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Definition 1.50 and §5.2.2, printed pp. 34 and 222
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.2.1 and the Jordan-form computation of spectra, printed pp. 219–222
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — the root-test step of Theorem 5.20, printed p. 222
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Exercise 5.4 and §5.1.1, printed pp. 209–213
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — §2.1, printed pp. 19–24
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Exercise 5.4 and §5.2.1, printed pp. 209–213 and 219–222
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — §2.1 and §2.3, printed pp. 19–24 and 30–33
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 5.20 and §5.1.1, printed pp. 209–214 and 222–223
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Lemmas 5.7–5.9 and Definition 5.8, printed pp. 213–216
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — §2.5, printed pp. 43–47
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Lemma 5.9 and §5.1.2, printed pp. 213–216
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Lemma 5.11 and Theorem 5.25(i), printed pp. 217–219 and 228
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Definition 5.24 and its cycle-existence remark, printed pp. 227–228
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Definition 5.24, printed pp. 227–228
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Definition 2.5.1, printed pp. 46–47
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 5.25(i), printed pp. 228–229
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — the contour-independence discussion after Definition 2.5.1, printed p. 47
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 5.25(ii)–(iii), printed pp. 228–230
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Theorem 2.5.2 and Exercise 2.5.3, printed pp. 47–48
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 5.25(iv)–(v), printed pp. 228 and 230–232
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Theorem 2.5.5, printed pp. 49–50
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 5.25(vi) and equation (5.26), printed pp. 226–228
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Exercise 2.5.3 and §2.5, printed pp. 48–50
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — Theorem 5.25(vi), printed p. 228
- Vahid Shirbisheh, Lectures on C-star Algebras, v2 — §2.5, printed pp. 48–50
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.1 (Banach algebras) used only for the quotient algebra conventions of this definition
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.1 (quotient algebra language; the Fredholm statement is the library's thm-atkinson, restated algebraically here)
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.2.1 (point, residual and continuous spectra), printed pp. 219–221
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.2.1 and Exercise 5.19-style presentation of the approximate point spectrum, printed pp. 219–221
- Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.2.1, printed pp. 219–221