Enclosure of the Numerical Range of a Class of Non-Selfadjoint Rational Operator Functions

TL;DR

This paper introduces a new, exactly computable enclosure of the numerical range for a class of rational operator functions, applicable in both finite and infinite dimensions, and useful for estimating the resolvent norm.

Contribution

It presents a novel enclosure that is minimal and computable, extending the concept of the numerical range to a broader class of operator functions.

Findings

Enclosure can be computed exactly in infinite dimensions

Provides a computable upper bound for the resolvent norm

Enclosure is minimal given the numerical ranges of coefficients

Abstract

In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.