The C-Numerical Range in Infinite Dimensions

Abstract

ABSTRACTIn infinite dimensions and on the level of trace-class operators C rather than matrices, we show that the closure of the C-numerical range is always star-shaped with respect to the set , where denotes the essential numerical range of the bounded operator T. Moreover, the closure of 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.