Pseudo Numerical Ranges and Spectral Enclosures

TL;DR

This paper introduces new concepts of pseudo numerical ranges for operator functions and matrices, providing spectral inclusion properties and bounds applicable to unbounded operators and damping in wave equations.

Contribution

It develops the theory of pseudo numerical ranges for unbounded operator functions and matrices, extending spectral enclosure results beyond traditional dominance conditions.

Findings

Spectral inclusion properties for pseudo numerical ranges

Spectral enclosures for operator matrices without dominance constraints

New spectral bounds for damped wave equations

Abstract

We introduce the new concepts of pseu\-do numerical range for operator functions and families of sesquilinear forms as well as the pseu\-do block numerical range for $n \times n$ operator matrix functions. While these notions are new even in the bounded case, we cover operator polynomials with unbounded coefficients, unbounded holomorphic form families of type (a) and associated operator families of type (B). Our main results include spectral inclusion properties of pseudo numerical ranges and pseudo block numerical ranges. For diagonally dominant and off-diagonally dominant operator matrices they allow us to prove spectral enclosures in terms of the pseudo numerical ranges of Schur complements that no longer require dominance order $0$ and not even $<1$. As an application, we establish a new type of spectral bounds for linearly damped wave equations with possibly unbounded and/or singular damping.