🎵Spectral Theory Unit 2 – Hilbert spaces and operators

Key Concepts and Definitions

• Hilbert space $H$ 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 $\|x\| = \sqrt{\langle x, x \rangle}$ 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 $x, y \in H$ are orthogonal if $\langle x, y \rangle = 0$
  • Orthonormal set is a collection of vectors that are pairwise orthogonal and have unit norm
• Completeness every Cauchy sequence in $H$ 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 $L^2(\mathbb{R})$ (square-integrable functions), $\ell^2$ (square-summable sequences), and $L^2([a,b])$ (square-integrable functions on a closed interval)
• Orthonormal bases an orthonormal set $\{e_n\}_{n=1}^\infty$ is an orthonormal basis for $H$ if every $x \in H$ can be uniquely represented as $x = \sum_{n=1}^\infty \langle x, e_n \rangle e_n$
  • Generalizes the concept of a basis from finite-dimensional vector spaces
• Parseval's identity for any $x \in H$ with orthonormal basis $\{e_n\}_{n=1}^\infty$, we have $\|x\|^2 = \sum_{n=1}^\infty |\langle x, e_n \rangle|^2$
  • Relates the norm of a vector to the sum of the squares of its Fourier coefficients
• Riesz representation theorem for every bounded linear functional $\varphi$ on $H$, there exists a unique $y \in H$ such that $\varphi(x) = \langle x, y \rangle$ for all $x \in H$
  • Establishes a correspondence between bounded linear functionals and vectors in the Hilbert space
• Orthogonal projection $P_M$ onto a closed subspace $M \subset H$ is a linear operator that satisfies $P_M^2 = P_M = P_M^*$ and $\text{Im}(P_M) = M$
  • Decomposes a vector into its components parallel and orthogonal to the subspace

Linear Operators on Hilbert Spaces

• Linear operator $T: H_1 \to H_2$ between Hilbert spaces satisfies $T(\alpha x + \beta y) = \alpha T(x) + \beta T(y)$ for all $x, y \in H_1$ and $\alpha, \beta \in \mathbb{C}$
• Bounded linear operator $T$ satisfies $\|Tx\| \leq C\|x\|$ for all $x \in H_1$ and some constant $C > 0$
  • Equivalently, $T$ is continuous or has a finite operator norm $\|T\| = \sup_{\|x\|=1} \|Tx\|$
• Adjoint operator $T^*: H_2 \to H_1$ satisfies $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x \in H_1$ and $y \in H_2$
  • 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 $\sigma(T)$ of a bounded linear operator $T$ is the set of $\lambda \in \mathbb{C}$ for which $T - \lambda I$ is not invertible
  • Consists of eigenvalues and approximate eigenvalues (if $T$ is not compact)

Spectral Properties of Operators

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

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., $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in H$
  • Spectrum of a self-adjoint operator is real and consists of the point spectrum and continuous spectrum
• Positive operator $T$ satisfies $\langle Tx, x \rangle \geq 0$ for all $x \in H$
  • Spectrum of a positive operator is a subset of $[0, \infty)$
• Square root $T^{1/2}$ of a positive operator $T$ is the unique positive operator satisfying $(T^{1/2})^2 = T$
  • 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 $T = \sum_{n=1}^\infty \lambda_n \langle \cdot, e_n \rangle e_n$, where $\{\lambda_n\}$ are the eigenvalues and $\{e_n\}$ the corresponding eigenvectors

Spectral Theorem and Applications

• Spectral theorem for bounded self-adjoint operators guarantees the existence of a unique projection-valued measure $E$ on the Borel subsets of $\mathbb{R}$ such that $T = \int_\mathbb{R} \lambda dE(\lambda)$
  • 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 $f(T) = \int_\mathbb{R} f(\lambda) dE(\lambda)$ 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 $\mathbb{C}$ such that $T = \int_\mathbb{C} \lambda dE(\lambda)$
  • 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 $L^2([0,1])$ with inner product $\langle f, g \rangle = \int_0^1 f(x)\overline{g(x)} dx$
  • Orthonormal basis $\{e^{2\pi i n x}\}_{n \in \mathbb{Z}}$ (complex exponentials)
• Example of a bounded linear operator multiplication operator $Mf(x) = xf(x)$ on $L^2([0,1])$
  • Self-adjoint with spectrum $\sigma(M) = [0,1]$ (continuous spectrum only)
• Example of a compact operator integral operator $Kf(x) = \int_0^1 k(x,y)f(y) dy$ on $L^2([0,1])$ with continuous kernel $k$
  • Compact, self-adjoint if $k(x,y) = \overline{k(y,x)}$, eigenvalues and eigenfunctions given by the Hilbert-Schmidt theorem
• Problem-solving strategy for determining the spectrum identify the point spectrum (eigenvalues) by solving $(T - \lambda I)v = 0$, then determine the continuous and residual spectra by analyzing the injectivity, surjectivity, and range of $T - \lambda I$
• 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 $D(T) \subset H$, closed if their graph $\{(x, Tx) : x \in D(T)\}$ is closed in $H \times H$
• Closed operators have a well-defined adjoint $T^*$ with domain $D(T^*) = \{y \in H : x \mapsto \langle Tx, y \rangle \text{ is bounded on } D(T)\}$
  • Spectrum and resolvent set can be defined for closed operators
• Self-adjoint extensions of a densely defined symmetric operator $T$ (satisfying $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in D(T)$) are self-adjoint operators $\tilde{T}$ such that $T \subset \tilde{T}$ and $D(T) \subset D(\tilde{T})$
  • 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 $H^2$ (a closed subspace of $L^2$ 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