A spectral radius for matrices over an operator space
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911052045746176 |
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| author | Shalit, Orr Shamovich, Eli |
| author_facet | Shalit, Orr Shamovich, Eli |
| contents | With every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$, we associate a spectral radius function $ρ_{\mathcal{E}}$ 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 $ρ_{\mathcal{E}}(X)<1$ 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 $ρ_{\mathcal{E}}$ is related to the joint spectral radius considered by Rota and Strang but differs from it and has the advantage that $ρ_{\mathcal{E}}(X)<1$ 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 $ρ_{\mathcal{E}}(A)<1$ if and only the domain of $f$ contains a neighborhood of the noncommutative closed unit ball of the operator space dual $\mathcal{E}^*$ of $\mathcal{E}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A spectral radius for matrices over an operator space Shalit, Orr Shamovich, Eli Operator Algebras Functional Analysis 47A13, 46L52, 47A10, 15A22 With every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$, we associate a spectral radius function $ρ_{\mathcal{E}}$ 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 $ρ_{\mathcal{E}}(X)<1$ 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 $ρ_{\mathcal{E}}$ is related to the joint spectral radius considered by Rota and Strang but differs from it and has the advantage that $ρ_{\mathcal{E}}(X)<1$ 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 $ρ_{\mathcal{E}}(A)<1$ if and only the domain of $f$ contains a neighborhood of the noncommutative closed unit ball of the operator space dual $\mathcal{E}^*$ of $\mathcal{E}$. |
| title | A spectral radius for matrices over an operator space |
| topic | Operator Algebras Functional Analysis 47A13, 46L52, 47A10, 15A22 |
| url | https://arxiv.org/abs/2501.01325 |