Bounded Linear Operators - Examples and Constructions

The spectrum of an operator generalizes the concept of eigenvalues from finite-dimensional linear algebra and plays a central role in the study of operators on Banach spaces.

DefinitionSpectrum and Resolvent

Let $T \in \mathcal{B}(X)$ where $X$ is a Banach space. A complex number $\lambda \in \mathbb{C}$ is in the resolvent set $\rho(T)$ if $(T - \lambda I)$ is bijective with bounded inverse.

The spectrum of $T$ is $\sigma(T) = \mathbb{C} \setminus \rho(T)$ .

For $\lambda \in \rho(T)$ , the operator $R_\lambda(T) = (T - \lambda I)^{-1}$ is called the resolvent of $T$ at $\lambda$ .

The spectrum decomposes into three disjoint parts:

Point Spectrum: $\sigma_p(T) = \{\lambda : T - \lambda I \text{ is not injective}\}$ (eigenvalues)
Continuous Spectrum: $\sigma_c(T) = \{\lambda : T - \lambda I \text{ is injective with dense range but not surjective}\}$
Residual Spectrum: $\sigma_r(T) = \{\lambda : T - \lambda I \text{ is injective but range is not dense}\}$

TheoremBasic Properties of the Spectrum

Let $T \in \mathcal{B}(X)$ where $X$ is a Banach space. Then:

The spectrum $\sigma(T)$ is a non-empty compact subset of $\mathbb{C}$
$\sigma(T) \subset \{\lambda \in \mathbb{C} : |\lambda| \leq \|T\|\}$
The resolvent set $\rho(T)$ is open and $R_\lambda(T)$ is analytic on $\rho(T)$
The spectral radius $r(T) = \sup\{|\lambda| : \lambda \in \sigma(T)\} = \lim_{n \to \infty} \|T^n\|^{1/n}$

ExampleSpectra of Standard Operators

Finite Dimensions: For $T : \mathbb{C}^n \to \mathbb{C}^n$ , we have $\sigma(T) = \sigma_p(T)$ (only eigenvalues)
Shift Operator: On $\ell^2$ , if $(Sx)_n = x_{n+1}$ , then:
- $\sigma(S) = \{\lambda : |\lambda| \leq 1\}$ (closed unit disk)
- $\sigma_p(S) = \emptyset$ (no eigenvalues)
- $\sigma_c(S) = \{\lambda : |\lambda| < 1\}$ (open disk)
- $\sigma_r(S) = \{\lambda : |\lambda| = 1\}$ (unit circle)
Multiplication Operator: On $L^2[0,1]$ , if $(M_\phi f)(x) = \phi(x) f(x)$ where $\phi$ is continuous, then $\sigma(M_\phi) = \overline{\text{Range}(\phi)}$ (closure of the range of $\phi$ )
Identity: $\sigma(I) = \{1\}$ and $\sigma(0) = \{0\}$

Remark

In infinite dimensions, the spectrum can be much richer than just eigenvalues. Operators may have no eigenvalues at all (like the shift operator), or the eigenvalues may form only a small part of the spectrum. This is one of the fundamental differences between finite and infinite-dimensional operator theory.

Understanding the spectrum is crucial for solving differential equations, analyzing stability of dynamical systems, and developing spectral methods for numerical approximation.