🎭Operator Theory Unit 5 – Spectral Theory: Bounded Self-Adjoint Operators

Key Concepts and Definitions

• Self-adjoint operators are linear operators $T$ on a Hilbert space $H$ satisfying $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y \in H$
• Bounded operators have finite operator norm $\|T\| = \sup_{\|x\|=1} \|Tx\|$
• Spectrum of an operator $T$ denoted by $\sigma(T)$ consists of all $\lambda \in \mathbb{C}$ such that $T - \lambda I$ is not invertible
  • Point spectrum $\sigma_p(T)$ contains eigenvalues of $T$
  • Continuous spectrum $\sigma_c(T)$ and residual spectrum $\sigma_r(T)$ are other subsets of $\sigma(T)$
• Resolvent set $\rho(T) = \mathbb{C} \setminus \sigma(T)$ is the complement of the spectrum
• Spectral radius $r(T) = \sup\{|\lambda| : \lambda \in \sigma(T)\}$ measures the size of the spectrum

Bounded Self-Adjoint Operators: Basics

• Self-adjoint operators are always normal operators satisfying $T^*T = TT^*$
• Real part $\Re(T) = \frac{1}{2}(T + T^*)$ and imaginary part $\Im(T) = \frac{1}{2i}(T - T^*)$ of an operator $T$ are self-adjoint
• Spectrum of a self-adjoint operator is always a subset of $\mathbb{R}$
• Eigenvalues of a self-adjoint operator are real, and eigenvectors corresponding to distinct eigenvalues are orthogonal
• Positive operators $T$ satisfy $\langle Tx, x \rangle \geq 0$ for all $x \in H$ and have non-negative spectrum
• Square root of a positive operator $T$ is the unique positive operator $S$ such that $S^2 = T$

Spectral Theorem for Bounded Self-Adjoint Operators

• Spectral theorem states that every bounded self-adjoint operator $T$ on a Hilbert space $H$ can be represented as an integral with respect to a unique spectral measure $E$
  • $T = \int_{\sigma(T)} \lambda dE(\lambda)$
• Spectral measure $E$ is a projection-valued measure on the Borel subsets of $\sigma(T)$
• For any Borel function $f: \sigma(T) \to \mathbb{C}$, the operator $f(T)$ is defined by $f(T) = \int_{\sigma(T)} f(\lambda) dE(\lambda)$
• Spectral theorem allows for the diagonalization of self-adjoint operators in an abstract sense
• Finite-dimensional case reduces to the existence of an orthonormal basis of eigenvectors for self-adjoint matrices

Continuous Functional Calculus

• Continuous functional calculus extends the notion of applying functions to self-adjoint operators
• For a continuous function $f: \sigma(T) \to \mathbb{C}$ and a self-adjoint operator $T$, the operator $f(T)$ is defined using the spectral theorem
• Functional calculus preserves algebraic operations: $(f + g)(T) = f(T) + g(T)$ and $(fg)(T) = f(T)g(T)$
• Composition of functions corresponds to the composition of operators: $(f \circ g)(T) = f(g(T))$
• Continuous functional calculus is a powerful tool for studying the properties of self-adjoint operators
  • Example: $e^{iT}$ is a unitary operator for any self-adjoint operator $T$

Applications in Quantum Mechanics

• Self-adjoint operators play a crucial role in quantum mechanics as observables
• Observables are physical quantities that can be measured, such as position, momentum, and energy
• Spectral theorem ensures that the spectrum of an observable corresponds to the possible outcomes of a measurement
• Eigenvectors of an observable represent the states in which the system has a definite value for that observable
• Time evolution of a quantum system is described by a unitary operator $U(t) = e^{-iHt}$, where $H$ is the Hamiltonian (energy) operator
• Functional calculus allows for the construction of functions of observables, such as the time evolution operator

Spectral Measures and Projections

• Spectral measure $E$ associated with a self-adjoint operator $T$ is a projection-valued measure on the Borel subsets of $\sigma(T)$
• For each Borel set $B \subseteq \sigma(T)$, $E(B)$ is a projection operator on a closed subspace of $H$
• Spectral projections $E(B)$ satisfy the properties of a measure:
  • $E(\emptyset) = 0$ and $E(\sigma(T)) = I$
  • For disjoint Borel sets $B_1, B_2, \ldots$, $E(\bigcup_{n=1}^{\infty} B_n) = \sum_{n=1}^{\infty} E(B_n)$
• Spectral projections allow for the decomposition of the Hilbert space $H$ into orthogonal subspaces corresponding to different parts of the spectrum
• Spectral measure can be used to express the expectation value of a self-adjoint operator $T$ in a state $x \in H$ as $\langle Tx, x \rangle = \int_{\sigma(T)} \lambda d\langle E(\lambda)x, x \rangle$

Examples and Problem-Solving Techniques

• Multiplication operator $Mf(x) = xf(x)$ on $L^2([a, b])$ is a bounded self-adjoint operator with spectrum $\sigma(M) = [a, b]$
• Laplace operator $\Delta = -\frac{d^2}{dx^2}$ on $L^2([0, 1])$ with Dirichlet boundary conditions is an unbounded self-adjoint operator with discrete spectrum $\sigma(\Delta) = \{n^2\pi^2 : n \in \mathbb{N}\}$
• To find the spectrum of a self-adjoint operator, consider the resolvent $(T - \lambda I)^{-1}$ and identify values of $\lambda$ for which it does not exist or is unbounded
• Spectral mapping theorem: for a continuous function $f: \sigma(T) \to \mathbb{C}$, $\sigma(f(T)) = f(\sigma(T))$
• Variational characterization of eigenvalues: for a self-adjoint operator $T$, the smallest eigenvalue $\lambda_1$ satisfies $\lambda_1 = \inf_{x \neq 0} \frac{\langle Tx, x \rangle}{\langle x, x \rangle}$

Advanced Topics and Extensions

• Unbounded self-adjoint operators can be defined on a dense subspace of a Hilbert space and have a more intricate spectral theory
• Spectral theorem for unbounded self-adjoint operators involves a more general notion of spectral measure and requires careful domain considerations
• Functional calculus can be extended to unbounded self-adjoint operators using the theory of Borel functions
• Spectral theory of self-adjoint operators can be generalized to the context of von Neumann algebras and non-commutative measure theory
• Spectral theory has applications in various areas of mathematics, including partial differential equations, harmonic analysis, and operator algebras
• Extensions of spectral theory include the study of normal operators, unitary operators, and more general classes of operators on Hilbert spaces