Resolvent identity - Imported Knowlpedia Content

Definition

Let $T$ be a closed linear operator on a complex Banach space, and write $R_T(\lambda)=(\lambda I-T)^{-1}$ for $\lambda$ in its resolvent set. The resolvent identity for two parameters $\lambda,\mu\in\rho(T)$ is

R_T(\lambda)-R_T(\mu) =(\mu-\lambda)R_T(\lambda)R_T(\mu).

It follows by inserting the factors $\lambda I-T$ and $\mu I-T$ and is an identity of bounded operators on the ambient space. In particular, the two resolvents commute. The name also covers the closely related identity comparing the resolvents of two different operators.

Analytic consequences

Taking $\mu$ close to $\lambda$ rewrites the identity as

R_T(\mu) =R_T(\lambda)\bigl(I-(\mu-\lambda)R_T(\lambda)\bigr)^{-1}.

The Neumann series for the inverse proves locally that the resolvent set is open and that $\lambda\mapsto R_T(\lambda)$ is holomorphic in operator norm. Differentiating with this sign convention gives

\frac{d}{d\lambda}R_T(\lambda)=-R_T(\lambda)^2.

These conclusions and their higher-derivative versions are standard tools in spectral perturbation theory Kato, Chapter III, §6.

Comparison of two operators

Suppose $A$ and $B$ are closed operators and $A-B$ acts on the range needed below, as happens when they have a common domain and $A-B$ extends to a bounded operator. At a common resolvent point $\lambda$ ,

R_A(\lambda)-R_B(\lambda) =R_A(\lambda)(A-B)R_B(\lambda).

This second resolvent identity converts control of the perturbation $A-B$ into control of the resolvents. Domain compatibility is essential for unbounded operators; the displayed product is not meaningful merely because $\lambda$ belongs to both resolvent sets.

Sign conventions

Some authors define the resolvent by $(T-\lambda I)^{-1}$ rather than $(\lambda I-T)^{-1}$ . The corresponding formulas have different displayed signs but identical mathematical content. “First” commonly refers to the two-parameter identity for one operator, while “second” refers to the comparison identity, although this terminology is not completely uniform.

References

Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1995. DOI record. Relevant: Chapter III, especially §6 on resolvents and spectra.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Publisher record. Relevant: Chapter VII on operator spectra and resolvents.