The C-Numerical Range in Infinite Dimensions

TL;DR

This paper extends the concept of the numerical range to infinite-dimensional trace-class operators, revealing geometric properties like star-shapedness and convexity under specific conditions, and explores the relationship between the $C$-spectrum and the $C$-numerical range.

Contribution

It introduces the $C$-numerical range in infinite dimensions, establishing its geometric properties and spectral relationships, which were previously understood mainly in finite-dimensional contexts.

Findings

The closure of the $C$-numerical range is star-shaped with respect to $ ext{tr}(C)W_e(T)$.

The closure of the $C$-numerical range is convex under certain conditions.

The $C$-spectrum is contained within the $C$-numerical range for compact normal operators.

Abstract

In infinite dimensions and on the level of trace-class operators $C$ rather than matrices, we show that the closure of the $C$-numerical range $W_C(T)$ is always star-shaped with respect to the set $\operatorname{tr}(C)W_e(T)$, where $W_e(T)$ denotes the essential numerical range of the bounded operator $T$. Moreover, the closure of $W_C(T)$ is convex if either $C$ is normal with collinear eigenvalues or if $T$ is essentially self-adjoint. In the case of compact normal operators, the $C$-spectrum of $T$ is a subset of the $C$-numerical range, which itself is a subset of the convex hull of the closure of the $C$-spectrum. This convex hull coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.