Resolvent of an element in a Banach algebra - Imported Knowlpedia Content

Definition

Let $A$ be a complex unital Banach algebra with identity $1$ , and let $a\in A$ . The resolvent set of $a$ is

\rho_A(a)=\{\lambda\in\mathbb C:\lambda1-a\text{ is invertible in }A\} =\mathbb C\setminus \sigma_A(a).

For $\lambda\in\rho_A(a)$ , the resolvent of $a$ at $\lambda$ is the inverse

R(\lambda,a)=(\lambda1-a)^{-1}.

Thus “resolvent” may denote either the $A$ -valued function $R(\,\cdot\,,a)$ or one of its values. For a nonunital Banach algebra, both spectrum and resolvent are defined using its unitization.

Analytic structure

The set $\rho_A(a)$ is open, and $R(\lambda,a)$ is holomorphic there. For $\lambda,\mu\in\rho_A(a)$ , the resolvent identity is

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

In particular, $R'(\lambda,a)=-R(\lambda,a)^2$ . If $|\lambda|>\lVert a\rVert$ , the Neumann series

R(\lambda,a)=\sum_{n=0}^{\infty}\lambda^{-n-1}a^n

converges in norm. These facts are established in Murphy, §1.2.

Estimates and example

The resolvent detects approach to the spectrum through

\lVert R(\lambda,a)\rVert\geq \frac{1}{\operatorname{dist}(\lambda,\sigma_A(a))}.

For a complex matrix $A$ , $\rho(A)$ is the complement of its eigenvalues and $R(\lambda,A)=(\lambda I-A)^{-1}$ . Near a nonnormal matrix’s spectrum, the norm of this inverse can be much larger than the reciprocal-distance lower bound; the resolvent therefore records more than the spectral set alone.

Conventions and scope

Some operator-theory texts define the resolvent as $(a-\lambda1)^{-1}$ , which differs by a minus sign from the convention used here. The identity and derivative formulas change signs accordingly. The ambient algebra matters: an element can become invertible in a larger algebra, changing both its spectrum and resolvent.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: Chapter 1, §1.2 on spectra and resolvents.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. DOI record. Relevant: Chapters 1–2 on inverses, spectra, and resolvent functions.