The C-Numerical Range for Schatten-Class Operators

Gunther Dirr, Frederik vom Ende

Published 2018-08-21Version 1

We generalize the $C$-numerical range $W_C(T)$ from trace-class to conjugate Schatten-class operators and show that its closure is always star-shaped with star-center $\lbrace 0\rbrace$. Equivalently, the closure of the image of the unitary orbit of $T\in\mathcal B^p(\mathcal H)$ under any continous linear functional $L\in(\mathcal B^p)'(\mathcal H)$ is star-shaped with star-center $\lbrace 0\rbrace$ if $p\in(1,\infty]$ and star-center $\operatorname{tr}(T)W_e(L)$ if $p=1$, where $W_e(L)$ denotes the essential numerical range of $L$. Moreover, the closure of $W_C(T)$ is convex if either $C$ or $T$ is normal with collinear eigenvalues. 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 closure of the convex hull of the $C$-spectrum. This closure coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.