Essential numerical range and $C$-numerical range for unbounded operators
Volume 264 / 2022
Abstract
We introduce two new concepts for unbounded operators $T$ in a Hilbert space, the essential numerical range $W_{e5}(T)$ of type $5$ and the $C$-numerical range $W_C(T)$. Our first main result clarifies the relation of $W_{e5}(T)$ to the essential numerical range $W_e(T)$, answering an open problem of Bögli, Marletta and Tretter’s (2020) by employing the Bessaga–Pełczyński selection theorem from Banach space theory. It turns out that $W_{e5}(T) \subset W_e(T)$ and we establish sharp conditions for equality. An example for strict inclusion shows that $W_e(T)$ may be a half-plane, while $W_{e5}(T)$ only a line. We also show that $W_{e5}(T)$ is convex and that it contains the convex hull of the essential spectrum. Our second main result reveals a geometric relation between $W_{e5}(T)$ and $\overline {W_C(T)}$. We show that, for finite-rank operators $C$, $\overline {W_C(T)}$ is star-shaped with star-centre $(\operatorname {Tr} C) W_{e5}(T)$, generalizing a result for bounded operators where $W_{e5}(T)=W_e(T)$.
