Point, continuous, and residual spectrum - Imported Knowlpedia Content

Definition

Let $T:\mathcal D(T)\subseteq X\to X$ be a densely defined operator on a complex Banach space, and assume that it is closed. A scalar $\lambda$ in the spectrum belongs to the point spectrum when $T-\lambda I$ is not injective, equivalently when $\lambda$ is an eigenvalue. If $T-\lambda I$ is injective, then $\lambda$ belongs to the continuous spectrum when its range is dense but not all of $X$ , and to the residual spectrum when its range is not dense. With this disjoint convention, these three sets partition the spectrum of a closed operator.

Hilbert-space description

For a densely defined operator on a complex Hilbert space,

\overline{\operatorname{Ran}(T-\lambda I)} =\ker(T^*-\overline{\lambda}I)^\perp.

Consequently, an injective $T-\lambda I$ has non-dense range exactly when $\overline{\lambda}$ is an eigenvalue of the adjoint $T^*$ . A self-adjoint operator has no residual spectrum: its spectrum is the disjoint union of its point and continuous spectra.

Representative examples

In finite-dimensional spaces every spectral value is an eigenvalue, so the continuous and residual spectra are empty. Multiplication by the coordinate function on $L^2([0,1])$ has spectrum $[0,1]$ and no eigenvalues; every point of the interval lies in its continuous spectrum. The unilateral shift

S(x_0,x_1,\ldots)=(0,x_0,x_1,\ldots)

on $\ell^2(\mathbb N)$ is injective, but its range is the closed proper subspace of sequences whose first coordinate is zero. Thus $0$ lies in its residual spectrum.

Conventions and scope

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Publisher record. Relevant: Chapter VII on the point, continuous, and residual spectra.
Nelson Dunford and Jacob T. Schwartz, Linear Operators, Part II: Spectral Theory, Interscience, 1963. Publisher record. Relevant: Chapter VII on spectral subdivisions.