A spectral radius for matrices over an operator space

TL;DR

This paper introduces a spectral radius function for operator space structures on matrices, characterizing when tuples are similar to strict contractions and connecting to existing spectral radius concepts.

Contribution

It defines a new spectral radius for operator space structures and relates it to similarity to strict contractions, extending classical spectral radius notions.

Findings

Spectral radius characterizes similarity to strict contractions.

Connection to joint spectral radius and existing operator space concepts.

Application to noncommutative rational functions and their domains.

Abstract

With every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$, we associate a spectral radius function $\rho_{\mathcal{E}}$ on $d$-tuples of operators. For a $d$-tuple $X = (X_1, \ldots, X_d) \in M_n(\mathbb{C}^d)$ of matrices we show that $\rho_{\mathcal{E}}(X)<1$ if and only if $X$ is jointly similar to a tuple in the open unit ball of $M_n(\mathcal{E})$, that is, there is an invertible matrix $S$ such that $\|S^{-1}X S\|_{M_n(\mathcal{E})}<1$, where $S^{-1} X S =(S^{-1} X_1 S, \ldots, S^{-1} X_d S)$. When $\mathcal{E}$ is the row operator space, for example, our spectral radius coincides with the joint spectral radius considered by Bunce, Popescu, and others, and we recover the condition for a tuple of matrices to be simultaneously similar to a strict row contraction. When $\mathcal{E}$ is the minimal operator space $\min(\ell^\infty_d)$, our spectral radius $\rho_{\mathcal{E}}$ is related to the joint spectral radius considered by Rota and Strang but differs from it and has the advantage that $\rho_{\mathcal{E}}(X)<1$ if and only if $X$ is simultaneously similar to a tuple of strict contractions. We show that for a nc rational function $f$ with descriptor realization $(A,b,c)$, the spectral radius $\rho_{\mathcal{E}}(A)<1$ if and only the domain of $f$ contains a neighborhood of the noncommutative closed unit ball of the operator space dual $\mathcal{E}^*$ of $\mathcal{E}$.