The C-Numerical Range in Infinite Dimensions

Frederik vom Ende, Gunther Dirr

Published 2017-12-04Version 1

In infinite dimensions, on the level of trace-class operators rather than matrices $C$, we show that the closure of the $C$-numerical range $\overline{W_C(T)}$ is 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 $T$. Further, we will see that 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 said $C$-spectrum. In addition, if the eigenvalues of $C$ are collinear, then the latter coincides with the closure of the $C$-numerical range for any compact normal operator $T$.