Spectral Theory

🎵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 HH is a complete inner product space over the field of real or complex numbers
  • Inner product ,\langle \cdot, \cdot \rangle is a generalization of the dot product that satisfies conjugate symmetry, linearity in the second argument, and positive definiteness
  • Norm x=x,x\|x\| = \sqrt{\langle x, x \rangle} measures the length or magnitude of a vector xx in the Hilbert space
    • Induced by the inner product and satisfies the triangle inequality and homogeneity
  • Orthogonality two vectors x,yHx, y \in H are orthogonal if x,y=0\langle x, y \rangle = 0
    • Orthonormal set is a collection of vectors that are pairwise orthogonal and have unit norm
  • Completeness every Cauchy sequence in HH 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)L^2(\mathbb{R}) (square-integrable functions), 2\ell^2 (square-summable sequences), and L2([a,b])L^2([a,b]) (square-integrable functions on a closed interval)
  • Orthonormal bases an orthonormal set {en}n=1\{e_n\}_{n=1}^\infty is an orthonormal basis for HH if every xHx \in H can be uniquely represented as x=n=1x,enenx = \sum_{n=1}^\infty \langle x, e_n \rangle e_n
    • Generalizes the concept of a basis from finite-dimensional vector spaces
  • Parseval's identity for any xHx \in H with orthonormal basis {en}n=1\{e_n\}_{n=1}^\infty, we have x2=n=1x,en2\|x\|^2 = \sum_{n=1}^\infty |\langle x, e_n \rangle|^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\varphi on HH, there exists a unique yHy \in H such that φ(x)=x,y\varphi(x) = \langle x, y \rangle for all xHx \in H
    • Establishes a correspondence between bounded linear functionals and vectors in the Hilbert space
  • Orthogonal projection PMP_M onto a closed subspace MHM \subset H is a linear operator that satisfies PM2=PM=PMP_M^2 = P_M = P_M^* and Im(PM)=M\text{Im}(P_M) = M
    • Decomposes a vector into its components parallel and orthogonal to the subspace

Linear Operators on Hilbert Spaces

  • Linear operator T:H1H2T: H_1 \to H_2 between Hilbert spaces satisfies T(αx+βy)=αT(x)+βT(y)T(\alpha x + \beta y) = \alpha T(x) + \beta T(y) for all x,yH1x, y \in H_1 and α,βC\alpha, \beta \in \mathbb{C}
  • Bounded linear operator TT satisfies TxCx\|Tx\| \leq C\|x\| for all xH1x \in H_1 and some constant C>0C > 0
    • Equivalently, TT is continuous or has a finite operator norm T=supx=1Tx\|T\| = \sup_{\|x\|=1} \|Tx\|
  • Adjoint operator T:H2H1T^*: H_2 \to H_1 satisfies Tx,y=x,Ty\langle Tx, y \rangle = \langle x, T^*y \rangle for all xH1x \in H_1 and yH2y \in H_2
    • Generalizes the concept of the transpose matrix in finite dimensions
  • Unitary operator UU satisfies UU=UU=IU^*U = UU^* = I, preserving inner products and norms
    • Isometric isomorphism between Hilbert spaces
  • Normal operator TT satisfies TT=TTTT^* = T^*T, commuting with its adjoint
    • Includes self-adjoint (T=TT = T^*), unitary, and orthogonal projection operators as special cases
  • Spectrum σ(T)\sigma(T) of a bounded linear operator TT is the set of λC\lambda \in \mathbb{C} for which TλIT - \lambda I is not invertible
    • Consists of eigenvalues and approximate eigenvalues (if TT is not compact)

Spectral Properties of Operators

  • Eigenvalue λC\lambda \in \mathbb{C} and corresponding eigenvector v0v \neq 0 satisfy Tv=λvTv = \lambda v for a linear operator TT
    • Eigenvectors corresponding to distinct eigenvalues are linearly independent
  • Eigenspace EλE_\lambda associated with an eigenvalue λ\lambda is the nullspace of TλIT - \lambda I
    • Closed subspace of HH consisting of all eigenvectors corresponding to λ\lambda (and the zero vector)
  • Point spectrum σp(T)\sigma_p(T) is the set of all eigenvalues of TT
    • Subset of the spectrum σ(T)\sigma(T)
  • Continuous spectrum σc(T)\sigma_c(T) is the set of λσ(T)\lambda \in \sigma(T) such that TλIT - \lambda I is injective with dense range, but not surjective
    • Corresponds to approximate eigenvectors and improper eigenfunctions
  • Residual spectrum σr(T)\sigma_r(T) is the set of λσ(T)\lambda \in \sigma(T) such that TλIT - \lambda I is injective but has non-dense range
    • Rarely encountered in practice and often empty for common operators
  • Spectral radius r(T)=sup{λ:λσ(T)}r(T) = \sup\{|\lambda| : \lambda \in \sigma(T)\} measures the size of the spectrum
    • Satisfies r(T)Tr(T) \leq \|T\| and r(Tn)=r(T)nr(T^n) = r(T)^n for all nNn \in \mathbb{N}
  • Resolvent set ρ(T)=Cσ(T)\rho(T) = \mathbb{C} \setminus \sigma(T) consists of all λC\lambda \in \mathbb{C} for which TλIT - \lambda I is invertible
    • Resolvent operator Rλ(T)=(TλI)1R_\lambda(T) = (T - \lambda I)^{-1} is a bounded linear operator for λρ(T)\lambda \in \rho(T)

Compact and Self-Adjoint Operators

  • Compact operator TT maps bounded sets to relatively compact sets (sets with compact closure)
    • Equivalently, TT 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 TT satisfies T=TT = T^*, i.e., Tx,y=x,Ty\langle Tx, y \rangle = \langle x, Ty \rangle for all x,yHx, y \in H
    • Spectrum of a self-adjoint operator is real and consists of the point spectrum and continuous spectrum
  • Positive operator TT satisfies Tx,x0\langle Tx, x \rangle \geq 0 for all xHx \in H
    • Spectrum of a positive operator is a subset of [0,)[0, \infty)
  • Square root T1/2T^{1/2} of a positive operator TT is the unique positive operator satisfying (T1/2)2=T(T^{1/2})^2 = T
    • Obtained via the functional calculus for self-adjoint operators
  • Polar decomposition T=UTT = U|T| for a bounded linear operator TT, where UU is a partial isometry and T=(TT)1/2|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=1λn,enenT = \sum_{n=1}^\infty \lambda_n \langle \cdot, e_n \rangle e_n, where {λn}\{\lambda_n\} are the eigenvalues and {en}\{e_n\} the corresponding eigenvectors

Spectral Theorem and Applications

  • Spectral theorem for bounded self-adjoint operators guarantees the existence of a unique projection-valued measure EE on the Borel subsets of R\mathbb{R} such that T=RλdE(λ)T = \int_\mathbb{R} \lambda dE(\lambda)
    • 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)f(T) = \int_\mathbb{R} f(\lambda) dE(\lambda) for any bounded Borel measurable function ff
    • 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 EE on the Borel subsets of C\mathbb{C} such that T=CλdE(λ)T = \int_\mathbb{C} \lambda dE(\lambda)
    • 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])L^2([0,1]) with inner product f,g=01f(x)g(x)dx\langle f, g \rangle = \int_0^1 f(x)\overline{g(x)} dx
    • Orthonormal basis {e2πinx}nZ\{e^{2\pi i n x}\}_{n \in \mathbb{Z}} (complex exponentials)
  • Example of a bounded linear operator multiplication operator Mf(x)=xf(x)Mf(x) = xf(x) on L2([0,1])L^2([0,1])
    • Self-adjoint with spectrum σ(M)=[0,1]\sigma(M) = [0,1] (continuous spectrum only)
  • Example of a compact operator integral operator Kf(x)=01k(x,y)f(y)dyKf(x) = \int_0^1 k(x,y)f(y) dy on L2([0,1])L^2([0,1]) with continuous kernel kk
    • Compact, self-adjoint if k(x,y)=k(y,x)k(x,y) = \overline{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 (TλI)v=0(T - \lambda I)v = 0, then determine the continuous and residual spectra by analyzing the injectivity, surjectivity, and range of TλIT - \lambda I
  • 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)HD(T) \subset H, closed if their graph {(x,Tx):xD(T)}\{(x, Tx) : x \in D(T)\} is closed in H×HH \times H
  • Closed operators have a well-defined adjoint TT^* with domain D(T)={yH:xTx,y is bounded on D(T)}D(T^*) = \{y \in H : x \mapsto \langle Tx, y \rangle \text{ is bounded on } D(T)\}
    • Spectrum and resolvent set can be defined for closed operators
  • Self-adjoint extensions of a densely defined symmetric operator TT (satisfying Tx,y=x,Ty\langle Tx, y \rangle = \langle x, Ty \rangle for all x,yD(T)x, y \in D(T)) are self-adjoint operators T~\tilde{T} such that TT~T \subset \tilde{T} and D(T)D(T~)D(T) \subset D(\tilde{T})
    • Existence and uniqueness of self-adjoint extensions characterized by the deficiency indices of TT
  • 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
  • CC^*-algebras and von Neumann algebras provide an abstract framework for studying algebras of bounded linear operators on Hilbert spaces
    • Gelfand-Naimark theorem characterizes CC^*-algebras as norm-closed self-adjoint subalgebras of B(H)B(H)
  • Toeplitz operators compressions of multiplication operators to the Hardy space H2H^2 (a closed subspace of L2L^2 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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.