Convexity and Star-shapedness of Matricial Range

TL;DR

This paper investigates the geometric properties of joint matricial ranges of operator tuples, proving star-shapedness in large dimensions and convexity of essential ranges, extending classical numerical range concepts.

Contribution

It establishes star-shapedness of the joint $(p,q)$-matricial range for large Hilbert spaces and introduces the convex, non-empty essential range for infinite-dimensional cases.

Findings

$oldsymbol{ ext{Lambda}}_{p,q}(oldsymbol{A})$ is star-shaped if the Hilbert space dimension is large.

The essential $(p,q)$-matricial range is convex, non-empty, and compact.

Extension of the range definition to infinite-dimensional spaces.

Abstract

Let ${\bf A} = (A_1, \dots, A_m)$ be an $m$-tuple of bounded linear operators acting on a Hilbert space ${\cal H}$. Their joint $(p,q)$-matricial range $\Lambda_{p,q}({\bf A})$ is the collection of $(B_1, \dots, B_m) \in {\bf M}_q^m$, 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 $\Lambda_{p,q}({\bf A})$ is star-shaped if the dimension of $\cal H$ is sufficiently large. If $\dim {\cal H}$ is infinite, we extend the definition of $\Lambda_{p,q}({\bf A})$ to $\Lambda_{\infty,q}({\bf A})$ consisting of $(B_1, \dots, B_m) \in {\bf M}_q^m$ such that $I_\infty \otimes B_j$ is a compression of $A_j$ on a closed subspace of ${\cal H}$, and consider the joint essential $(p,q)$-matricial range $$\Lambda^{ess}_{p,q}({\bf A}) = \bigcap \{ {\bf cl}(\Lambda_{p,q}(A_1+F_1, \dots, A_m+F_m)): F_1, \dots, F_m \hbox{ are compact operators}\}.$$ Both sets are shown to be convex, and the latter one is always non-empty and compact.