The CNumerical Range in Infinite Dimensions
Abstract
In infinite dimensions and on the level of traceclass operators $C$ rather than matrices, we show that the closure of the $C$numerical range $W_C(T)$ is always starshaped with respect to the set $\operatorname{tr}(C)W_e(T)$, 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 selfadjoint. 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.01023
 arXiv:
 arXiv:1712.01023
 Bibcode:
 2017arXiv171201023D
 Keywords:

 Mathematics  Functional Analysis;
 Mathematical Physics;
 47A12 (Primary) 15A60 (Secondary)
 EPrint:
 31 pages, no figures