The C-Numerical Range in Infinite Dimensions

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 $\mathrm{tr}(C)W_e(T)$, 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.

31 pages, no figures; to appear in Linear and Multilinear Algebra

References in corpus (1)

• The Significance of the $C$-Numerical Range and the Local $C$-Numerical Range in Quantum Control and Quantum Information

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• Convexity of the orbit-closed $C$-numerical range and majorization