On the spectral radius of operator tuples

TL;DR

This paper investigates how the spectral radius of operator tuples depends on the underlying operator space structure, revealing that for commuting tuples it depends only on the normed space, and exploring conditions for equality of spectral radii across different spaces.

Contribution

It establishes the relationship between spectral radius and operator space structures, characterizes spectral radius in terms of invertibility domains, and compares spectral radii for different operator spaces.

Findings

Spectral radius depends only on the underlying normed space for commuting tuples.

For dimensions ≥ 3, spectral radii differ for minimal and maximal operator space structures.

Equality of spectral radii for all tuples implies the operator spaces are identical.

Abstract

In recent work, Shalit and Shamovich associated to every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ a spectral radius function $\rho_{\mathcal{E}}$ on $d$-tuples of operators. The main goal of this paper is to elucidate how this spectral radius depends on the operator space structure. Let $V = (\mathbb{C}^d, \|\cdot\|_V)$ be a normed space and let $\mathcal{E}$ be a quantization of $V$. We show that for a commuting operator tuple $X$, the spectral radius depends only on the underlying normed space; more precisely, \[ \rho_{\mathcal{E}}(X) = \max\{ \|\lambda\|_V : \lambda \in \sigma(X)\}, \] where $\sigma(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V \geq 3$, then $\rho_{\min(V)}(X) \neq \rho_{\max(V)}(X)$ already for some matrix tuple $X$. When $\mathcal{E}_1$ and $\mathcal{E}_2$ are selfadjoint operator spaces, we show that $\rho_{\mathcal{E}_1}(X) = \rho_{\mathcal{E}_2}(X)$ for all tuples $X$ implies $\mathcal{E}_1 = \mathcal{E}_2$. We present two proofs of this result; a key ingredient in one of them is a characterization, of independent interest, of $\rho_{\mathcal{E}}(A)$ in terms of the invertibility domain of the linear pencil associated with $A$. Finally, we prove that if two operator spaces give rise to the same spectral radius function, then the algebras of locally uniformly bounded NC functions on the corresponding NC unit balls coincide.