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 $\ddot{I}{}_{\mathcal{E}}$ 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 $V=({\mathbb{C}}^d,\Vert \cdot {\Vert }_V)$ 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, \[ ρ_{\mathcal{E}}(X) = \max\{ \|λ\|_V : λ\in σ(X)\}, \] where $\ddot{I}ƒ(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V\ge 3$, then $\ddot{I}{}_{\min (V)}(X)\ne \ddot{I}{}_{\max (V)}(X)$ already for some matrix tuple $X$. When ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ are selfadjoint operator spaces, we show that $\ddot{I}{}_{{\mathcal{E}}_1}(X)=\ddot{I}{}_{{\mathcal{E}}_2}(X)$ 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 $\ddot{I}{}_{\mathcal{E}}(A)$ 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.