The C-Numerical Range for Schatten-Class Operators

We generalize the $C$-numerical range $W_C(T)$ from trace-class to Schatten-class operators, i.e. to $C\in\mathcal B^p(\mathcal H)$ and $T\in\mathcal B^q(\mathcal H)$ with $1/p + 1/q = 1$, and show that its closure is always star-shaped with respect to the origin. For $q \in (1,\infty]$, this is equivalent to saying that the closure of the image of the unitary orbit of $T\in\mathcal B^q(\mathcal H)$ under any continous linear functional $L\in(\mathcal B^q(\mathcal H))'$ is star-shaped with respect to the origin. For $q=1$, one has star-shapedness with respect to $\operatorname{tr}(T)W_e(L)$, where $W_e(L)$ denotes the essential range of $L$. Moreover, the closure of $W_C(T)$ is convex if $C$ or $T$ is normal with collinear eigenvalues. If $C$ and $T$ are both normal, then 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.