Spectral Theory of Operators — Mathematics Knowledge System

Overview

Spectral theory extends the notion of eigenvalues and eigenvectors from finite-dimensional linear algebra to operators on infinite-dimensional Banach and Hilbert spaces. The spectrum of an operator $T$ replaces the finite set of eigenvalues and decomposes into the point spectrum (eigenvalues), continuous spectrum, and residual spectrum. For bounded self-adjoint operators on Hilbert spaces, the spectral theorem provides a complete decomposition analogous to diagonalisation: every such operator can be realised as a multiplication operator on an $L^2$ space, encoded by a projection-valued measure.

Intuition

In finite dimensions, every symmetric matrix can be diagonalised. Spectral theory asks: can we diagonalise an infinite-dimensional operator? For compact self-adjoint operators the answer is an exact analogue of the finite case—eigenvalues accumulate only at 0, and eigenvectors form a complete orthonormal basis. For general bounded self-adjoint operators, diagonalisation requires a continuum of generalised eigenvectors encoded by a spectral measure. The spectrum is the set of values at which the operator almost fails to be invertible.

Formal Definition

Definition

Let $T \in \mathcal{B}(X)$ be a bounded linear operator on a complex Banach space $X$. The resolvent set is $\rho(T) = \{\lambda \in \mathbb{C} : (T - \lambda I)^{-1} \in \mathcal{B}(X)\}$. The spectrum is $\sigma(T) = \mathbb{C} \setminus \rho(T)$. The spectrum decomposes as: point spectrum $\sigma_p(T)$ where $T - \lambda I$ is not injective; continuous spectrum $\sigma_c(T)$ where $T - \lambda I$ is injective with dense but not closed range; residual spectrum $\sigma_r(T)$ where $T - \lambda I$ is injective but the range is not dense.

\sigma(T) = \{\lambda \in \mathbb{C} : (T - \lambda I) \text{ is not boundedly invertible}\}

Spectrum

\sigma(T) = \sigma_p(T) \cup \sigma_c(T) \cup \sigma_r(T) \quad (\text{disjoint})

Spectral decomposition

r(T) = \sup_{\lambda \in \sigma(T)} |\lambda| = \lim_{n\to\infty} \|T^n\|^{1/n}

Spectral radius formula

Properties

Spectrum of self-adjoint operator is real

If T \in \mathcal{B}(H) is self-adjoint (T^* = T), then \sigma(T) \subset \mathbb{R}.

Spectrum of unitary operator lies on unit circle

If U is unitary (U^*U = UU^* = I), then \sigma(U) \subset \{\lambda : |\lambda| = 1\}.

Eigenvectors for distinct eigenvalues of self-adjoint operators are orthogonal

If Te_1 = \lambda_1 e_1 and Te_2 = \lambda_2 e_2 with \lambda_1 \neq \lambda_2 and T self-adjoint, then \langle e_1, e_2 \rangle = 0.

Theorems

Theorem 5.1: Spectral Radius Formula

For T \in \mathcal{B}(X), the spectral radius satisfies r(T) = \lim_{n\to\infty} \|T^n\|^{1/n} \le \|T\|.

Theorem 5.2: Spectrum is compact and nonempty

For any T \in \mathcal{B}(X) on a complex Banach space, \sigma(T) is a nonempty compact subset of \{\lambda : |\lambda| \le \|T\|\}.

Theorem 5.3: Spectral Theorem for Bounded Self-Adjoint Operators

If T is a bounded self-adjoint operator on a Hilbert space H, then there exists a unique projection-valued measure E on \sigma(T) \subset \mathbb{R} such that T = \int_{\sigma(T)} \lambda\,dE(\lambda).

Theorem 5.4: Spectral Theorem for Compact Self-Adjoint Operators

If T is a compact self-adjoint operator on a separable Hilbert space H, then H has an orthonormal basis \{e_n\} of eigenvectors of T with corresponding real eigenvalues \lambda_n \to 0.

Worked Examples

Point spectrum: $\lambda \in \sigma_p(M_\phi)$ iff $\phi(x) = \lambda$ on a set of positive measure. Since $\{x = \lambda\}$ has measure zero for every $\lambda$, $\sigma_p(M_\phi) = \emptyset$.
Resolvent: $(M_\phi - \lambda I)^{-1}$ acts by multiplication by $1/(x-\lambda)$. This is bounded iff $1/(x-\lambda) \in L^\infty([0,1])$, which fails when $\lambda \in [0,1]$.
$\|(M_\phi - \lambda I)^{-1}\| = \text{ess-sup}_{x \in [0,1]} \frac{1}{|x - \lambda|}$
For $\lambda \notin [0,1]$, $1/(x-\lambda)$ is bounded on $[0,1]$, so $\lambda \in \rho(M_\phi)$. Hence $\sigma(M_\phi) = [0,1]$.
Since $M_\phi - \lambda I$ is injective for all $\lambda \in [0,1]$ (multiplication by $x-\lambda$ is injective on $L^2$) but not boundedly invertible, the entire interval $[0,1]$ is in $\sigma_c(M_\phi)$.

Answer: $\sigma(M_x) = [0,1]$ with purely continuous spectrum; no eigenvalues.

Practice Problems

Difficulty 8/10

Prove that the spectrum of a bounded operator on a complex Banach space is nonempty.

Difficulty 9/10

Compute $\sigma(S)$ and $\sigma(S^*)$ where $S$ is the unilateral right shift on $\ell^2$: $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$.

Difficulty 9/10

Prove that eigenvectors of a self-adjoint operator corresponding to distinct eigenvalues are orthogonal.

Common Mistakes

Common Mistake

The spectrum consists only of eigenvalues

In infinite dimensions the spectrum has three disjoint parts: point spectrum (eigenvalues), continuous spectrum, and residual spectrum. The multiplication operator $M_x$ on $L^2([0,1])$ has spectrum $[0,1]$ but no eigenvalues.

Common Mistake

Spectral radius always equals the operator norm

The spectral radius $r(T) = \lim \|T^n\|^{1/n} \le \|T\|$; equality holds for normal operators ($T^*T = TT^*$) but fails for non-normal ones. The nilpotent matrix $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ has norm 1 but spectral radius 0.

Quiz

Summary

The spectrum $\sigma(T)$ is the set of $\lambda$ for which $T - \lambda I$ fails to be boundedly invertible; it is always compact and nonempty for bounded operators on complex Banach spaces.
The spectrum decomposes into point spectrum (eigenvalues), continuous spectrum, and residual spectrum.
Spectral radius formula: $r(T) = \lim_n \|T^n\|^{1/n}$; for self-adjoint operators $r(T) = \|T\|$.
Self-adjoint operators on Hilbert spaces have real spectrum and admit the spectral theorem: $T = \int \lambda\,dE(\lambda)$.
Compact self-adjoint operators have a countable spectrum accumulating only at 0, with eigenvectors forming a complete orthonormal basis.

References

BookRudin, W. Functional Analysis. 2nd ed., McGraw-Hill, 1991. Chapters 10–12.
BookReed, M. and Simon, B. Methods of Modern Mathematical Physics, Vol. 1. Academic Press, 1980.
WebsiteWikipedia — Spectral theory