Convexity and Star-shapedness of Matricial Range

Abstract

Let $\mathbf{A}=(A_1,\dots ,A_m)$ be an $m$-tuple of bounded linear operators acting on a Hilbert space $\mathcal{H}$. Their joint $(p,q)$-matricial range ${\Lambda }_{p,q}(\mathbf{A})$ is the collection of $(B_1,\dots ,B_m)\in {\mathbf{M}}_q^m$, where $I_p\otimes B_j$ is a compression of $A_j$ on a $pq$-dimensional subspace. This definition covers various kinds of generalized numerical ranges for different values of $p,q,m$. In this paper, it is shown that ${\Lambda }_{p,q}(\mathbf{A})$ is star-shaped if the dimension of $\mathcal{H}$ is sufficiently large. If $\dim \mathcal{H}$ is infinite, we extend the definition of ${\Lambda }_{p,q}(\mathbf{A})$ to ${\Lambda }_{\infty ,q}(\mathbf{A})$ consisting of $(B_1,\dots ,B_m)\in {\mathbf{M}}_q^m$ such that $I_{\infty }\otimes B_j$ is a compression of $A_j$ on a closed subspace of $\mathcal{H}$, and consider the joint essential $(p,q)$-matricial range $${\Lambda }_{p,q}^{ess}(\mathbf{A})=\bigcap \{\mathbf{cl}({\Lambda }_{p,q}(A_1+F_1,\dots ,A_m+F_m)):F_1,\dots ,F_{m}are\; compact\; operators\}.$$ Both sets are shown to be convex, and the latter one is always non-empty and compact.

14 pages