Preservation of the joint essential matricial range

TL;DR

This paper characterizes the preservation of the joint essential matricial range under compact perturbations and establishes its relation to the Calkin algebra, extending previous results to multiple operators and matrix sizes.

Contribution

It generalizes known results about the essential numerical range to the joint matricial setting for multiple operators and matrix sizes, linking it to the Calkin algebra.

Findings

Proves that the joint essential spatial q-matricial range equals the joint q-matricial range of the image in the Calkin algebra.

Shows existence of compact operators making the closure of the spatial q-matricial range equal to the essential range for all q up to N.

Extends previous results by Narcowich-Ward, Smith-Ward, and Müller to the joint setting and higher matrix sizes.

Abstract

Let $A = (A_1, \dots, A_m)$ be an $m$-tuple of elements of a unital $C$*-algebra ${\cal A}$ and let $M_q$ denote the set of $q \times q$ complex matrices. The joint $q$-matricial range $W^q(A)$ is the set of $(B_1, \dots, B_m) \in M_q^m$ such that $B_j = \Phi(A_j)$ for some unital completely positive linear map $\Phi: {\cal A} \rightarrow M_q$. When ${\cal A}= B(H)$, where $B(H)$ is the algebra of bounded linear operators on the Hilbert space $H$, the {\bf joint spatial $q$-matricial range} $W^q_s(A)$ of $A$ is the set of $(B_1, \dots, B_m) \in M_q^m$ for which there is a $q$-dimensional $V$ of $H$ such that $B_j$ is a compression of $A_j$ to $V$ for $j=1,\dots, m$. Suppose $K(H)$ is the set of compact operators in $B(H)$. The joint essential spatial $q$-matricial range is defined as $$W_{ess}^q(A) = \cap \{ {\bf cl}(W_s^q(A_1+K_1, \dots, A_m+K_m)): K_1, \dots, K_m \in K(H) \},$$ where ${\bf cl}$ denotes the closure. Let $\pi$ be the canonical surjection from $B(H)$ to the Calkin algebra $B(H)/K(H)$. We prove that $W_{ess}^q(A) =W^q(\pi(A) $, where $\pi(A) = (\pi(A_1), \dots, \pi(A_m))$. Furthermore, for any positive integer $N$, we prove that there are self-adjoint compact operators $K_1, \dots, K_m$ such that $${\bf cl}(W^q_s(A_1+K_1, \dots, A_m+K_m)) = W^q_{ess}(A) \quad \hbox{ for all } q \in \{1, \dots, N\}.$$ These results generalize those of Narcowich-Ward and Smith-Ward, obtained in the $m=1$ case, and also generalize a result of M\"{u}ller obtained in case $m \ge 1$ and $q=1$. Furthermore, if $W_{ess}^1({\bf A}) $ is a simplex in ${\mathbb R}^m$, then we prove that there are self-adjoint $K_1, \dots, K_m \in K(H)$ such that ${\bf cl}(W^q_s(A_1+K_1, \dots, A_m+K_m)) = W^q_{ess}(A)$ for all positive integers $q$.