Operator with compact resolvent - Imported Knowlpedia Content

Definition

Let $T:\mathcal D(T)\subseteq X\to X$ be a densely defined closed operator on a Banach space, and suppose its resolvent set $\rho(T)$ is nonempty. The operator $T$ has compact resolvent if

(T-\lambda I)^{-1}:X\longrightarrow X

is a compact operator for some $\lambda\in\rho(T)$ . The resolvent identity then implies compactness for every $\lambda\in\rho(T)$ , so the definition does not depend on the chosen resolvent point. Compact resolvent is stronger than closedness: it says that solving $(T-\lambda I)x=y$ sends bounded sets of data to relatively compact sets in $X$ .

Compact domain embedding

Give $\mathcal D(T)$ the graph norm $\|x\|_T=\|x\|+\|Tx\|$ . When $\rho(T)\neq\varnothing$ , $T$ has compact resolvent exactly when the inclusion

(\mathcal D(T),\|\cdot\|_T)\hookrightarrow X

is compact. This reformulation often proves compactness through an embedding theorem rather than by constructing the resolvent explicitly.

Spectral consequences

Every point of the spectrum of a compact-resolvent operator is an isolated eigenvalue of finite algebraic multiplicity, and the spectrum has no finite accumulation point. The spectrum may be empty for a general non-self-adjoint unbounded operator. If it is infinite, its eigenvalues can accumulate only at infinity.

Self-adjoint case and examples

For a self-adjoint operator $D$ on a Hilbert space, compact resolvent is equivalent to compactness of $(1+D^2)^{-1/2}$ . The spectral theorem then provides an orthonormal basis of eigenvectors, with eigenvalues of finite multiplicity whose absolute values tend to infinity. Elliptic differential operators on compact manifolds, subject to suitable elliptic boundary conditions when a boundary is present, are standard examples.

References