标题： 算子空间上矩阵的谱半径 显示英文标题

标题： A spectral radius for matrices over an operator space

摘要： 对于 $\mathcal{E}$ 在 $\mathbb{C}^d$上的每个算子空间结构，我们关联一个谱半径函数 $\rho_{\mathcal{E}}$ 在 $d$-元组的算子上。 对于一个矩阵的$d$-元组$X = (X_1, \ldots, X_d) \in M_n(\mathbb{C}^d)$，我们证明了$\rho_{\mathcal{E}}(X)<1$当且仅当$X$在联合意义下与$M_n(\mathcal{E})$的开单位球中的某个元组相似，也就是说，存在一个可逆矩阵$S$使得$\|S^{-1}X S\|_{M_n(\mathcal{E})}<1$，其中$S^{-1} X S =(S^{-1} X_1 S, \ldots, S^{-1} X_d S)$。 当 $\mathcal{E}$ 是行算子空间时，例如，我们的谱半径与 Bunce、Popescu 等人所考虑的联合谱半径一致，并且我们恢复了矩阵组同时相似于严格行压缩的条件。 当 $\mathcal{E}$ 是最小算子空间 $\min(\ell^\infty_d)$ 时，我们的谱半径 $\rho_{\mathcal{E}}$ 与 Rota 和 Strang 考虑的联合谱半径有关但有所不同，并且具有优势：$\rho_{\mathcal{E}}(X)<1$ 当且仅当 $X$ 同时相似于一个严格压缩元组。 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 $\mathcal{E}$. 显示英文摘要

摘要： 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 $\mathcal{E}$.