6.5 Spectral representation theorem

The spectral representation theorem is a cornerstone of operator theory, providing a powerful framework for analyzing linear operators on Hilbert spaces. It connects abstract algebraic properties to concrete geometric structures, enabling deep insights into operator behavior.

This theorem establishes an isomorphism between self-adjoint operators and multiplication operators, allowing for a decomposition of Hilbert spaces into spectral subspaces. It generalizes finite-dimensional diagonalization to infinite-dimensional settings, forming the foundation for many applications in physics and engineering.

Foundations of spectral representation

• Spectral representation forms the cornerstone of operator theory in functional analysis
• Provides a powerful framework for analyzing linear operators on Hilbert spaces
• Connects abstract algebraic properties of operators to concrete geometric and analytic structures

Hilbert space basics

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• Complete inner product spaces generalize finite-dimensional vector spaces
• Orthogonality and projections play crucial roles in Hilbert space geometry
• Separable Hilbert spaces have countable orthonormal bases (Fourier series)
• Riesz representation theorem establishes duality between elements and continuous linear functionals

Linear operators overview

• Maps between Hilbert spaces preserve vector space structure
• Bounded operators have finite operator norms and are continuous
• Adjoint operators satisfy $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all vectors x and y
• Spectrum generalizes eigenvalues to infinite-dimensional settings
• Resolvent set consists of complex numbers where $(T - \lambda I)^{-1}$ exists as a bounded operator

Spectral theory fundamentals

• Studies properties of operators through their spectra
• Spectral radius formula: $r(T) = \lim_{n \to \infty} \|T^n\|^{1/n}$
• Gelfand's formula relates spectral radius to operator norm
• Spectral mapping theorem connects operator functions to spectrum transformations
• Fredholm alternative characterizes solvability of operator equations

Self-adjoint operators

• Central objects in spectral theory due to their symmetry properties
• Model many physical systems in quantum mechanics and other fields
• Spectral theorem for self-adjoint operators provides powerful decomposition

Definition and properties

• Satisfy $T = T^*$ where $T^*$ denotes the adjoint operator
• Have real spectrum: $\sigma(T) \subseteq \mathbb{R}$
• Norm equals spectral radius: $\|T\| = r(T)$
• Positive operators form important subclass (T ≥ 0 if $\langle Tx, x \rangle \geq 0$ for all x)
• Polar decomposition: every bounded operator factors as T = U|T| with U partial isometry

Bounded vs unbounded operators

• Bounded operators defined on entire Hilbert space, continuous everywhere
• Unbounded operators only defined on dense subspace, discontinuous at boundary
• Closed unbounded operators have closed graphs in product space
• Symmetric operators ($\langle Tx, y \rangle = \langle x, Ty \rangle$ on domain) may have self-adjoint extensions
• Cayley transform provides bijection between self-adjoint operators and certain unitary operators

Spectrum of self-adjoint operators

• Consists entirely of real numbers
• Divides into point spectrum, continuous spectrum, and residual spectrum
• Spectral theorem decomposes operator using projection-valued measure
• Essential spectrum persists under compact perturbations
• Weyl's criterion characterizes essential spectrum via singular sequences

Spectral measure

• Generalizes notion of projection-valued measure to operator-valued case
• Provides rigorous foundation for functional calculus of self-adjoint operators
• Connects spectral properties to measure theory and integration

Definition and construction

• Projection-valued measure on Borel subsets of spectrum
• Satisfies countable additivity in strong operator topology
• Constructed via Gelfand-Naimark-Segal (GNS) construction for C*-algebras
• Spectral family $\{E_\lambda\}_{\lambda \in \mathbb{R}}$ of projections defines spectral measure
• Resolution of identity: $I = \int_{\sigma(T)} dE(\lambda)$ where I denotes identity operator

Properties of spectral measure

• Support contained in spectrum of operator
• Commutes with operator and its functions
• Orthogonal projections: $E(A)E(B) = E(A \cap B)$ for Borel sets A, B
• Spectral measure of singleton $\{\lambda\}$ gives eigenprojection if λ is eigenvalue
• Lebesgue decomposition into absolutely continuous, singular continuous, and pure point parts

Borel functional calculus

• Extends continuous functional calculus to measurable functions
• For Borel function f, defines $f(T) = \int_{\sigma(T)} f(\lambda) dE(\lambda)$
• Preserves algebraic operations: $(fg)(T) = f(T)g(T)$
• Spectral mapping theorem: $\sigma(f(T)) = f(\sigma(T))$ (with exceptions for constant functions)
• Provides rigorous meaning to functions of unbounded self-adjoint operators

Spectral representation theorem

• Fundamental result in spectral theory of self-adjoint operators
• Establishes isomorphism between operator and multiplication operator
• Allows decomposition of Hilbert space into spectral subspaces

Statement of the theorem

• Every self-adjoint operator T on Hilbert space H unitarily equivalent to multiplication operator
• Exists unitary U: H → L²(X, μ) and measurable function m such that UTU⁻¹ = M_m
• M_m denotes multiplication by m: $(M_m f)(\lambda) = m(\lambda)f(\lambda)$
• Measure space (X, μ) and multiplier m determined by spectral measure of T
• Generalizes finite-dimensional diagonalization to infinite-dimensional setting

Proof outline and key steps

• Construct maximal abelian von Neumann algebra generated by T
• Apply Gelfand representation to obtain isomorphism with C(X) for compact X
• Use Riesz-Markov theorem to represent states as measures on X
• Construct spectral measure via GNS construction
• Show unitary equivalence between T and multiplication operator
• Handle unbounded case via Cayley transform or spectral family

Uniqueness of representation

• Representation unique up to unitary equivalence
• Multiplicity function determines decomposition into cyclic subspaces
• Spectral type (absolutely continuous, singular continuous, pure point) invariant
• Möbius inversion formula recovers spectral measure from resolvent
• Stone's formula expresses spectral projections in terms of resolvent

Spectral decomposition

• Decomposes Hilbert space into subspaces corresponding to different spectral types
• Provides finer structure than eigendecomposition in finite-dimensional case
• Crucial for understanding behavior of operator and its functions

Continuous spectrum

• Points λ where T - λI not invertible but has dense range
• Corresponds to absolutely continuous part of spectral measure
• Gives rise to continuous subspace H_c in direct sum decomposition
• Examples include multiplication operator on L²[0,1] and position operator in quantum mechanics
• Weyl-von Neumann theorem allows approximation by operators with pure point spectrum

Point spectrum

• Set of eigenvalues of operator T
• Corresponds to pure point part of spectral measure
• Gives rise to pure point subspace H_pp in direct sum decomposition
• Countable set for self-adjoint operators on separable Hilbert space
• Includes discrete spectrum (isolated eigenvalues of finite multiplicity)

Residual spectrum

• Empty for self-adjoint operators
• For normal operators, consists of points λ where T - λI not invertible and range not dense
• Important for studying non-self-adjoint operators (Volterra operator)
• Related to approximate point spectrum and compression spectrum
• Spectral mapping theorem may fail for residual spectrum

Applications of spectral representation

• Provides powerful tools for analyzing diverse physical and mathematical systems
• Connects abstract operator theory to concrete applications in science and engineering
• Enables solution of complex problems through spectral decomposition and functional calculus

Quantum mechanics

• Self-adjoint operators represent observables in quantum systems
• Spectral theorem gives physical interpretation to measurement outcomes
• Energy levels correspond to point spectrum of Hamiltonian operator
• Continuous spectrum describes scattering states in unbounded systems
• Time evolution governed by unitary groups via Stone's theorem

Signal processing

• Fourier transform as spectral decomposition of translation operator
• Windowed Fourier transform and wavelet transforms use generalized eigenfunctions
• Karhunen-Loève transform optimally decorrelates stochastic processes
• Filter design utilizes spectral properties of convolution operators
• Sampling theorems relate discrete and continuous spectral representations

Functional analysis

• Spectral theory unifies treatment of differential and integral operators
• Fredholm theory characterizes compact perturbations of identity
• Index theory connects spectral properties to topological invariants
• C*-algebras and von Neumann algebras extend spectral theory to noncommutative setting
• Spectral flow measures spectral changes in families of self-adjoint operators

Generalizations and extensions

• Broadens scope of spectral theory beyond self-adjoint operators
• Addresses more general classes of operators and spaces
• Connects spectral theory to other branches of mathematics and physics

Normal operators

• Commute with their adjoints: $TT^* = T^*T$
• Include self-adjoint, unitary, and multiplication operators
• Spectral theorem generalizes to normal operators via complex-valued spectral measure
• Fuglede-Putnam theorem relates commutation properties to spectral measures
• Aluthge transform provides bridge between normal and non-normal operators

Compact operators

• Generalize finite-rank operators to infinite-dimensional spaces
• Have discrete spectrum except possibly for 0
• Fredholm alternative characterizes solvability of equations involving compact operators
• Trace class and Hilbert-Schmidt operators form important subclasses
• Lidskii trace formula connects traces to eigenvalues for trace class operators

Spectral theorem for unbounded operators

• Extends spectral representation to unbounded self-adjoint operators
• Uses Cayley transform to relate unbounded operators to bounded ones
• Spectral family $\{E_\lambda\}_{\lambda \in \mathbb{R}}$ of projections defines operator
• Functional calculus extends to unbounded measurable functions
• Essential self-adjointness guarantees unique self-adjoint extension

Computational aspects

• Bridges theoretical spectral theory with practical numerical methods
• Enables application of spectral techniques to real-world problems
• Addresses challenges of infinite-dimensional operators in finite computational settings

Numerical methods for spectral analysis

• Finite element methods approximate spectra of differential operators
• Lanczos algorithm efficiently computes extremal eigenvalues of large matrices
• Arnoldi iteration generalizes Lanczos to non-Hermitian matrices
• Density functional theory uses spectral methods in electronic structure calculations
• Spectral collocation methods solve PDEs using global basis functions

Approximation techniques

• Galerkin methods project infinite-dimensional problems onto finite subspaces
• Weyl sequences approximate points in essential spectrum
• Finite section method truncates infinite matrices to approximate spectra
• Polynomial approximation of spectral projectors via contour integrals
• Padé approximants provide rational approximations to spectral functions

Software tools for spectral theory

• ARPACK implements implicitly restarted Arnoldi method for large eigenproblems
• SLEPc extends PETSc for scalable eigenvalue computations
• FEAST algorithm uses contour integration for interior eigenvalue problems
• TensorFlow and PyTorch enable spectral computations on GPUs for machine learning
• Chebfun system implements spectral methods in MATLAB for function approximation