Commuting normal operators and joint numerical range

TL;DR

This paper investigates the geometric and algebraic properties of joint numerical ranges of commuting operators on Hilbert spaces, characterizing when these ranges are polyhedral and exploring implications for normal and compact operators.

Contribution

It provides new characterizations of when joint numerical ranges are polyhedral, linking geometric properties to algebraic structures of operators, especially in the context of commuting and normal operators.

Findings

Existence of a finite-dimensional reducing subspace with diagonal compression for polyhedral joint numerical ranges.

Equivalence of conditions for commuting normal operators and polyhedral joint numerical ranges in finite rank case.

Differences in properties between finite rank and compact operators regarding joint numerical ranges.

Abstract

Let ${\mathcal H}$ be a complex Hilbert space and let ${\mathcal B}({\mathcal H})$ be the algebra of all bounded linear operators on ${\mathcal H}$. For a positive integer $k$ less than the dimension of ${\mathcal H}$ and ${\mathbf A} = (A_1, \dots, A_m)\in {\mathcal B}({\mathcal H})^m$, the joint $k$-numerical range $W_k({\mathbf A})$ is the set of $(\alpha_1, \dots, \alpha_m) \in{\mathbb C}^m$ such that $\alpha_i = \sum_{j = 1}^k \langle A_ix_j, x_j\rangle$ for an orthonormal set $\{x_1, \ldots, x_k\}$ in ${\mathcal H}$. Relations between the geometric properties of $W_k({\mathbf A})$ and the algebraic and analytic properties of $A_1, \dots, A_m$ are studied. It is shown that there is $k\in {\mathbb N}$ such that $W_k({\mathbf A})$ is a polyhedral set, i.e., the convex hull of a finite set, if and only if $A_1, \dots, A_k$ have a common reducing subspace ${\mathbf V}$ of finite dimension such that the compression of $A_1, \dots, A_m$ on the subspace ${\mathbf V}$ are diagonal operators $D_1, \dots, D_m$ and $W_k({\mathbf A}) = W_k(D_1, \dots, D_m)$. Characterization is also given to ${\bf A}$ such that the closure of $W_k({\mathbf A})$ is polyhedral. The conditions are related to the joint essential numerical range of ${\mathbf A}$. These results are used to study ${\bf A}$ such that (a) $\{A_1, \dots, A_m\}$ is a commuting family of normal operators, or (b) $W_k(A_1, \dots, A_m)$ is polyhedral for every positive integer $k$. It is shown that conditions (a) and (b) are equivalent for finite rank operators but it is no longer true for compact operators. Characterizations are given for compact operators $A_1, \dots, A_m$ satisfying (a) and (b), respectively. Results are also obtained for general non-compact operators.