The higher rank numerical range of matrix polynomials

TL;DR

This paper introduces the higher rank numerical range for matrix polynomials, explores its geometric properties, and investigates its relation to sharp points and vector-valued ranges, expanding the understanding of matrix polynomial spectra.

Contribution

It defines the higher rank numerical range for matrix polynomials and studies its fundamental geometric properties and relations to other spectral concepts.

Findings

Characterization of the higher rank numerical range $\Lambda_{k}(L(\lambda))$

Relation between sharp points of $\Lambda_{k}(L(\lambda))$ and the numerical range

Connection between $\Lambda_{k}(L(\lambda))$ and vector-valued higher rank numerical range

Abstract

The notion of the higher rank numerical range $\Lambda_{k}(L(\lambda))$ for matrix polynomials $L(\lambda)=A_{m}\lambda^{m}+...+A_{1}\lambda+A_{0}$ is introduced here and some fundamental geometrical properties are investigated. Further, the sharp points of $\Lambda_{k}(L(\lambda))$ are defined and their relation to the numerical range $w(L(\lambda))$ is presented. A connection of $\Lambda_{k}(L(\lambda))$ with the vector-valued higher rank numerical range $\Lambda_{k}(A_{0},..., A_{m})$ is also discussed.