Joint q-Numerical Ranges of Continuous Operator-Valued Functions

Abstract

This paper develops a unified theory of joint $q$-numerical ranges $\Fq(F)$ for continuous functions $F:\Om \to \B(\cH)$, where $\Om$ is a compact Hausdorff space and $\B(\cH)$ denotes the $C^*$-algebra of bounded linear operators on a complex Hilbert space $\cH$. We prove that $\Fq(F)$ is totally bounded in $\cC(\Om)$, the space of continuous complex-valued functions on $\Om$, and establish the upper semicontinuity of the set-valued map $F\mapsto \Fq(F)$, and show that $\Fq(F)$ is compact when $\cH$ is finite-dimensional. The main result of the paper is a realization theorem. We prove that, for every totally bounded subset $K$ of $\cC(\Om)$ and every $q\in \C$ with $|q|=1$, there exist a Hilbert space $\cH$ and a continuous function $F:\Om\to\B(\cH)$ such that $qK\subset \Fq(F)\subset q\cok$. Consequently, if $K$ is closed and convex, then $\Fq(F)=qK$. As applications, we derive analogous realization results for bounded subsets in $\C^n$ and for totally bounded subsets in arbitrary Banach spaces.