On the spectral radius of operator tuples

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.