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 WCβ(T) is
always star-shaped with respect to the set tr(C)Weβ(T), where
Weβ(T) denotes the essential numerical range of the bounded operator T.
Moreover, the closure of WCβ(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.Comment: 31 pages, no figures; to appear in Linear and Multilinear Algebr