A spectral radius for matrices over an operator space

Abstract

With every operator space structure $E$ on $C^d$, we associate a spectral radius function ${\rho }_E$ on d-tuples of operators. For a d-tuple $X=(X_1,\dots ,X_d)\in M_n\left(C^d\right)$ of matrices we show that ${\rho }_E\left(X\right)<1$ if and only if X is jointly similar to a tuple in the open unit ball of $M_n\left(E\right)$, that is, there is an invertible matrix S such that ${\Vert S^{-1}XS\Vert }_{M_n\left(E\right)}<1$, where $S^{-1}XS=(S^{-1}{X}_{1}S,\dots ,S^{-1}{X}_{d}S)$. More generally, for all $X\in B\left(K\right){\otimes }_{\min }E$ we show that ${\rho }_E\left(X\right)<1$ if and only if there exists an invertible $S\in B\left(K\right)\otimes I$ such that $\Vert S^{-1}XS\Vert <1$. When $E$ 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 $E$ is the minimal operator space $\min \left({\ell }_d^{\infty }\right)$, our spectral radius ${\rho }_E$ is related to the joint spectral radius considered by Rota and Strang but differs from it and has the advantage that ${\rho }_E\left(X\right)<1$ if and only if X is simultaneously similar to a tuple of strict contractions. We show that for an nc rational function f with descriptor realization $(A,b,c)$, the spectral radius ${\rho }_E\left(A\right)<1$ if and only if the domain of f contains a neighborhood of the noncommutative closed unit ball of the operator space dual $E^{⁎}$ of $E$.