Alphabeta MathContinuous Functional Calculus for Self Adjoint and Normal Operators
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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
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- 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
- Complex Norms and Realification
- Complex Subspace Realification
- Dual Spaces Adjoint Operators and Annihilators
- Gelfand Theory and Commutative C Star Algebras
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- 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
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- 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 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
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Lemma
Bounded Hilbert operators form a C star algebra
Lemma
Spectrum of a self adjoint operator is real
Lemma
Spectrum of a positive operator is nonnegative
Definition
Order on bounded self adjoint operators
Definition
C star algebra generated by a normal operator
Corollary
Normal operator norm equals spectral radius
Corollary
Normal operator with zero spectrum is zero
Definition
Isometry coisometry and partial isometry
Theorem
Partial isometry characterizations
Definition
Numerical range and numerical radius
Theorem
Numerical radius is an equivalent operator norm
Lemma
Polynomial calculus is isometric for self adjoint operators
Theorem
Continuous functional calculus for bounded self adjoint operators
Lemma
Spectral permanence for unital c star subalgebras
Lemma
Character space of generated normal algebra is operator spectrum
Theorem
Continuous functional calculus for bounded normal operators
Theorem
Spectral mapping for continuous normal functional calculus
Theorem
Continuous functional calculus properties
Theorem
Self adjoint norm and spectrum extrema
Theorem
Positive square root
Definition
Absolute value of a bounded operator
Theorem
Polar decomposition for bounded operators
Theorem
Bounded normal operator abstract spectral theorem
Remark
Positive square root and covariance matrices
Lemma
Two dimensional numerical range is convex
Theorem
Toeplitz hausdorff
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.3, printed pp.235–245
- Dana P. Williams, Lecture Notes on the Spectral Theorem, §3, pp.6–10
- Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.49, printed pp.238–240
- Dana P. Williams, Lecture Notes on the Spectral Theorem, §4, pp.10–13
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.4, printed pp.245–255
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.69, printed p.268
- John B. Conway, A Course in Functional Analysis, 2nd ed., Chapter IX §3, printed pp.239–243
- Joel H. Shapiro, Notes on the Numerical Range, §3, PDF pp.9–11
- Joel H. Shapiro, Notes on the Numerical Range, §3–4, PDF pp.9–12
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Corollary 4.11, pp.13–15
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.54, printed pp.250–262
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Example 4.9, pp.13–15
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Theorem 4.8, pp.13–15
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.69 and Theorem 5.70, printed pp.268–273
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.70, printed pp.268–273
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.54 and Theorem 5.70, printed pp.250–273
- Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.49 and §5.4, printed pp.238–255
- Dana P. Williams, Lecture Notes on the Spectral Theorem, §4, pp.10–15
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.54 and §5.4, printed pp.250–260
- John B. Conway, A Course in Functional Analysis, 2nd ed., Polar Decomposition 3.11, printed pp.239–243
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.4, printed pp.245–260
- Joel H. Shapiro, Notes on the Numerical Range, §5, PDF pp.11–15
- Joel H. Shapiro, Notes on the Numerical Range, Theorem 6.1, PDF pp.15–17