Spectral Theory - Core Definitions

Spectral theory generalizes eigenvalue decomposition from finite-dimensional linear algebra to infinite-dimensional operators, providing a powerful tool for analyzing differential and integral operators.

DefinitionSpectral Decomposition

A spectral measure for an operator $T$ on a Hilbert space $H$ is a projection-valued measure $E : \mathcal{B}(\mathbb{C}) \to \mathcal{B}(H)$ such that $T = \int_{\sigma(T)} \lambda \, dE(\lambda)$

This generalizes the eigenvalue decomposition $T = \sum \lambda_n \langle \cdot, e_n \rangle e_n$ for compact operators.

TheoremSpectral Theorem for Bounded Self-Adjoint Operators

Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$ . Then there exists a unique spectral measure $E$ on $\sigma(T) \subset \mathbb{R}$ such that $T = \int_{\sigma(T)} \lambda \, dE(\lambda)$

Moreover, for any continuous function $f : \sigma(T) \to \mathbb{C}$ , we can define $f(T) = \int_{\sigma(T)} f(\lambda) \, dE(\lambda)$

This theorem provides a functional calculus: we can apply functions to operators in a meaningful way.

ExampleSpectral Decompositions

Compact Self-Adjoint: For compact $T$ , the spectral measure is atomic: $E(B) = \sum_{\lambda_n \in B} \langle \cdot, e_n \rangle e_n$
Multiplication Operator: On $L^2(\Omega)$ , if $(M_\phi f)(x) = \phi(x) f(x)$ , then $M_\phi = \int_{\mathbb{R}} \lambda \, dE_\lambda$ where $E_\lambda$ projects onto $\{f : \phi^{-1}(-\infty, \lambda]\}$
Position and Momentum: In quantum mechanics, position $\hat{x}$ and momentum $\hat{p} = -i\hbar \frac{d}{dx}$ have continuous spectra

DefinitionFunctional Calculus

Given a bounded self-adjoint operator $T$ with spectral measure $E$ , we can define:

Polynomials: $p(T) = \sum a_n T^n$ in the usual sense
Continuous Functions: $f(T) = \int f(\lambda) \, dE(\lambda)$ for $f \in C(\sigma(T))$
Borel Functions: $f(T) = \int f(\lambda) \, dE(\lambda)$ for bounded Borel $f$

TheoremProperties of Functional Calculus

The functional calculus satisfies:

$(f + g)(T) = f(T) + g(T)$
$(fg)(T) = f(T)g(T)$
$\overline{f}(T) = f(T)^*$
$\|f(T)\| = \|f\|_\infty = \sup_{\lambda \in \sigma(T)} |f(\lambda)|$

Remark

The spectral theorem transforms the study of operators into the study of multiplication operators, which are much easier to understand. This principle underlies the solution of many PDEs, where differential operators are "diagonalized" via Fourier transform or eigenfunction expansions.

Spectral theory is the mathematical foundation for quantum mechanics, signal processing, and the numerical analysis of differential operators.