🎵Spectral Theory Unit 2 – Hilbert spaces and operators
Hilbert spaces and operators form the backbone of functional analysis, providing a powerful framework for studying infinite-dimensional vector spaces. This unit covers key concepts like inner products, orthogonality, and completeness, which generalize familiar ideas from finite-dimensional linear algebra.
Linear operators on Hilbert spaces are explored, including bounded, self-adjoint, and compact operators. The spectral theorem, a cornerstone result, allows for the decomposition of operators and has far-reaching applications in quantum mechanics, signal processing, and partial differential equations.
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Key Concepts and Definitions
Hilbert space H is a complete inner product space over the field of real or complex numbers
Inner product ⟨⋅,⋅⟩ is a generalization of the dot product that satisfies conjugate symmetry, linearity in the second argument, and positive definiteness
Norm ∥x∥=⟨x,x⟩ measures the length or magnitude of a vector x in the Hilbert space
Induced by the inner product and satisfies the triangle inequality and homogeneity
Orthogonality two vectors x,y∈H are orthogonal if ⟨x,y⟩=0
Orthonormal set is a collection of vectors that are pairwise orthogonal and have unit norm
Completeness every Cauchy sequence in H converges to a limit within the space
Ensures that the space contains all its limit points and has no "gaps"
Separability a Hilbert space is separable if it has a countable dense subset
Allows for approximation of elements by a countable set of basis vectors
Hilbert Space Fundamentals
Hilbert spaces generalize Euclidean spaces to infinite dimensions while preserving key properties like the inner product and completeness
Examples of Hilbert spaces include L2(R) (square-integrable functions), ℓ2 (square-summable sequences), and L2([a,b]) (square-integrable functions on a closed interval)
Orthonormal bases an orthonormal set {en}n=1∞ is an orthonormal basis for H if every x∈H can be uniquely represented as x=∑n=1∞⟨x,en⟩en
Generalizes the concept of a basis from finite-dimensional vector spaces
Parseval's identity for any x∈H with orthonormal basis {en}n=1∞, we have ∥x∥2=∑n=1∞∣⟨x,en⟩∣2
Relates the norm of a vector to the sum of the squares of its Fourier coefficients
Riesz representation theorem for every bounded linear functional φ on H, there exists a unique y∈H such that φ(x)=⟨x,y⟩ for all x∈H
Establishes a correspondence between bounded linear functionals and vectors in the Hilbert space
Orthogonal projection PM onto a closed subspace M⊂H is a linear operator that satisfies PM2=PM=PM∗ and Im(PM)=M
Decomposes a vector into its components parallel and orthogonal to the subspace
Linear Operators on Hilbert Spaces
Linear operator T:H1→H2 between Hilbert spaces satisfies T(αx+βy)=αT(x)+βT(y) for all x,y∈H1 and α,β∈C
Bounded linear operator T satisfies ∥Tx∥≤C∥x∥ for all x∈H1 and some constant C>0
Equivalently, T is continuous or has a finite operator norm ∥T∥=sup∥x∥=1∥Tx∥
Adjoint operator T∗:H2→H1 satisfies ⟨Tx,y⟩=⟨x,T∗y⟩ for all x∈H1 and y∈H2
Generalizes the concept of the transpose matrix in finite dimensions
Unitary operator U satisfies U∗U=UU∗=I, preserving inner products and norms
Isometric isomorphism between Hilbert spaces
Normal operator T satisfies TT∗=T∗T, commuting with its adjoint
Includes self-adjoint (T=T∗), unitary, and orthogonal projection operators as special cases
Spectrum σ(T) of a bounded linear operator T is the set of λ∈C for which T−λI is not invertible
Consists of eigenvalues and approximate eigenvalues (if T is not compact)
Spectral Properties of Operators
Eigenvalue λ∈C and corresponding eigenvector v=0 satisfy Tv=λv for a linear operator T
Eigenvectors corresponding to distinct eigenvalues are linearly independent
Eigenspace Eλ associated with an eigenvalue λ is the nullspace of T−λI
Closed subspace of H consisting of all eigenvectors corresponding to λ (and the zero vector)
Point spectrum σp(T) is the set of all eigenvalues of T
Subset of the spectrum σ(T)
Continuous spectrum σc(T) is the set of λ∈σ(T) such that T−λI is injective with dense range, but not surjective
Corresponds to approximate eigenvectors and improper eigenfunctions
Residual spectrum σr(T) is the set of λ∈σ(T) such that T−λI is injective but has non-dense range
Rarely encountered in practice and often empty for common operators
Spectral radius r(T)=sup{∣λ∣:λ∈σ(T)} measures the size of the spectrum
Satisfies r(T)≤∥T∥ and r(Tn)=r(T)n for all n∈N
Resolvent set ρ(T)=C∖σ(T) consists of all λ∈C for which T−λI is invertible
Resolvent operator Rλ(T)=(T−λI)−1 is a bounded linear operator for λ∈ρ(T)
Compact and Self-Adjoint Operators
Compact operator T maps bounded sets to relatively compact sets (sets with compact closure)
Equivalently, T is the norm limit of finite-rank operators or maps weakly convergent sequences to strongly convergent sequences
Spectrum of a compact operator consists only of eigenvalues (point spectrum) and 0
Eigenvalues have finite multiplicities (dimensions of corresponding eigenspaces) and can only accumulate at 0
Self-adjoint operator T satisfies T=T∗, i.e., ⟨Tx,y⟩=⟨x,Ty⟩ for all x,y∈H
Spectrum of a self-adjoint operator is real and consists of the point spectrum and continuous spectrum
Positive operator T satisfies ⟨Tx,x⟩≥0 for all x∈H
Spectrum of a positive operator is a subset of [0,∞)
Square root T1/2 of a positive operator T is the unique positive operator satisfying (T1/2)2=T
Obtained via the functional calculus for self-adjoint operators
Polar decomposition T=U∣T∣ for a bounded linear operator T, where U is a partial isometry and ∣T∣=(T∗T)1/2 is positive
Generalizes the polar form of a complex number to operators
Spectral theorem for compact self-adjoint operators guarantees the existence of an orthonormal basis of eigenvectors
Operator can be diagonalized as T=∑n=1∞λn⟨⋅,en⟩en, where {λn} are the eigenvalues and {en} the corresponding eigenvectors
Spectral Theorem and Applications
Spectral theorem for bounded self-adjoint operators guarantees the existence of a unique projection-valued measure E on the Borel subsets of R such that T=∫RλdE(λ)
Generalizes the eigendecomposition of compact self-adjoint operators to the non-compact case
Functional calculus for bounded self-adjoint operators allows for the definition of f(T)=∫Rf(λ)dE(λ) for any bounded Borel measurable function f
Consistent with the usual definition of polynomials and power series of operators
Spectral theorem for bounded normal operators guarantees the existence of a unique projection-valued measure E on the Borel subsets of C such that T=∫CλdE(λ)
Generalizes the spectral theorem for self-adjoint operators to the complex case
Application to quantum mechanics self-adjoint operators represent observable quantities, with their spectra corresponding to possible measurement outcomes
Spectral measures model the probabilistic nature of quantum measurements
Application to signal processing compact self-adjoint operators (e.g., integral operators with symmetric kernels) can be used to model linear time-invariant systems
Eigenvalues and eigenfunctions provide a frequency-domain representation of the system
Application to PDEs spectral theory can be used to solve linear PDEs by expanding solutions in terms of eigenfunctions of the associated differential operators
Leads to efficient numerical methods for solving PDEs on bounded domains
Examples and Problem-Solving Strategies
Example of a Hilbert space L2([0,1]) with inner product ⟨f,g⟩=∫01f(x)g(x)dx
Example of a bounded linear operator multiplication operator Mf(x)=xf(x) on L2([0,1])
Self-adjoint with spectrum σ(M)=[0,1] (continuous spectrum only)
Example of a compact operator integral operator Kf(x)=∫01k(x,y)f(y)dy on L2([0,1]) with continuous kernel k
Compact, self-adjoint if k(x,y)=k(y,x), eigenvalues and eigenfunctions given by the Hilbert-Schmidt theorem
Problem-solving strategy for determining the spectrum identify the point spectrum (eigenvalues) by solving (T−λI)v=0, then determine the continuous and residual spectra by analyzing the injectivity, surjectivity, and range of T−λI
Problem-solving strategy for diagonalization use the spectral theorem for compact self-adjoint operators or the functional calculus for bounded self-adjoint operators
Express the operator in terms of its eigenvalues and eigenvectors or spectral measure
Problem-solving strategy for approximation approximate a given function or operator by a sequence of simpler functions or operators (e.g., polynomials, finite-rank operators)
Use the properties of Hilbert spaces and operators to prove convergence and estimate the approximation error
Advanced Topics and Extensions
Unbounded operators linear operators that are not necessarily bounded or continuous
Defined on a dense subspace D(T)⊂H, closed if their graph {(x,Tx):x∈D(T)} is closed in H×H
Closed operators have a well-defined adjoint T∗ with domain D(T∗)={y∈H:x↦⟨Tx,y⟩ is bounded on D(T)}
Spectrum and resolvent set can be defined for closed operators
Self-adjoint extensions of a densely defined symmetric operator T (satisfying ⟨Tx,y⟩=⟨x,Ty⟩ for all x,y∈D(T)) are self-adjoint operators T~ such that T⊂T~ and D(T)⊂D(T~)
Existence and uniqueness of self-adjoint extensions characterized by the deficiency indices of T
Spectral theory of unbounded self-adjoint operators generalizes the spectral theorem and functional calculus to the unbounded case
Relies on the theory of Borel measures and Lebesgue integration
Banach space operators spectral theory can be extended to bounded linear operators on Banach spaces (complete normed vector spaces)
Spectrum and resolvent set can be defined, but the spectral theorem and functional calculus may not hold
C∗-algebras and von Neumann algebras provide an abstract framework for studying algebras of bounded linear operators on Hilbert spaces
Gelfand-Naimark theorem characterizes C∗-algebras as norm-closed self-adjoint subalgebras of B(H)
Toeplitz operators compressions of multiplication operators to the Hardy space H2 (a closed subspace of L2 on the unit circle)
Spectral properties and index theory relate to the symbol of the operator and the winding number of its determinant
Wavelet theory studies orthonormal bases and frames generated by dilations and translations of a single function (wavelet)
Provides efficient representations for signals and operators in various function spaces