Resolvent set of a closed operator - Imported Knowlpedia Content

Definition

Let $T:\operatorname{Dom}(T)\subseteq X\to X$ be a closed linear operator on a complex Banach space. Its resolvent set is

\rho(T)=\{\lambda\in\mathbb C:\lambda I-T\text{ is bijective and } (\lambda I-T)^{-1}:X\to X\text{ is bounded}\}.

For $\lambda\in\rho(T)$ , the bounded linear operator

R(\lambda,T)=(\lambda I-T)^{-1}

is the resolvent operator. Its range lies in $\operatorname{Dom}(T)$ , although it is viewed as an operator on $X$ . Because $T$ is closed, bijectivity already forces this inverse to be bounded by the closed graph theorem.

Basic properties

The resolvent set is open in $\mathbb C$ , and $\lambda\mapsto R(\lambda,T)$ is operator-norm holomorphic there. If $\lambda,\mu\in\rho(T)$ , then

R(\lambda,T)-R(\mu,T) =(\mu-\lambda)R(\lambda,T)R(\mu,T).

This resolvent identity gives a local Neumann-series expansion and explains both openness and holomorphy. The spectrum of $T$ is the complement $\mathbb C\setminus\rho(T)$ ; for an unbounded operator that complement need not be bounded.

Domain and sign conventions

Some authors define the resolvent as $(T-\lambda I)^{-1}$ , which differs from the convention above by a minus sign. Either convention gives the same set $\rho(T)$ . Boundedness here is measured in the ambient norm of $X$ , not only in the graph norm on $\operatorname{Dom}(T)$ . Closedness is what turns an algebraic inverse defined on all of $X$ into a bounded one Kato, Chapter III.

Example

For the multiplication operator $(Tf)(x)=x f(x)$ on $L^2(\mathbb R)$ , with maximal domain $\{f:xf\in L^2(\mathbb R)\}$ , every $\lambda\notin\mathbb R$ lies in $\rho(T)$ . The resolvent multiplies by $(\lambda-x)^{-1}$ , whose essential supremum is at most $|\operatorname{Im}\lambda|^{-1}$ . No real $\lambda$ is in the resolvent set.

References

Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1995. Publisher record. Relevant: Chapter III on closed operators, spectra, and resolvents.
Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012. Publisher record. Relevant: Chapter 1, §3 on the spectrum of a closed operator.