Enclosure of the Numerical Range and Resolvent Estimates of Non-selfadjoint Operator Functions

TL;DR

This paper explores the connection between the numerical range of unbounded operator functions and their coefficients, providing methods for explicit enclosures and resolvent estimates that are optimal given joint numerical range bounds.

Contribution

It introduces new techniques to estimate the numerical range and resolvent of non-selfadjoint operator functions based on joint numerical range analysis.

Findings

Derived explicit enclosures of the numerical range.

Provided optimal resolvent norm estimates.

Established methods for estimating joint numerical range.

Abstract

In this paper we discuss the relationship between the numerical range of an extensive class of unbounded operator functions and the joint numerical range of the operator coefficients. Furthermore, we derive methods on how to find estimates of the joint numerical range. Those estimates are used to obtain explicitly computable enclosures of the numerical range of the operator function and resolvent estimates. The enclosure and upper estimate of the norm of the resolvent are optimal given the estimate of the joint numerical range.