On the spectral radius of operator tuples
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915998639063040 |
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| author | Scherer, Marcel Shalit, Orr Shamovich, Eli |
| author_facet | Scherer, Marcel Shalit, Orr Shamovich, Eli |
| contents | In recent work, Shalit and Shamovich associated to every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ a spectral radius function $ρ_{\mathcal{E}}$ 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, \[ ρ_{\mathcal{E}}(X) = \max\{ \|λ\|_V : λ\in σ(X)\}, \] where $σ(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V \geq 3$, then $ρ_{\min(V)}(X) \neq ρ_{\max(V)}(X)$ already for some matrix tuple $X$. When $\mathcal{E}_1$ and $\mathcal{E}_2$ are selfadjoint operator spaces, we show that $ρ_{\mathcal{E}_1}(X) = ρ_{\mathcal{E}_2}(X)$ 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 $ρ_{\mathcal{E}}(A)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09354 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the spectral radius of operator tuples Scherer, Marcel Shalit, Orr Shamovich, Eli Operator Algebras Functional Analysis In recent work, Shalit and Shamovich associated to every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ a spectral radius function $ρ_{\mathcal{E}}$ 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, \[ ρ_{\mathcal{E}}(X) = \max\{ \|λ\|_V : λ\in σ(X)\}, \] where $σ(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V \geq 3$, then $ρ_{\min(V)}(X) \neq ρ_{\max(V)}(X)$ already for some matrix tuple $X$. When $\mathcal{E}_1$ and $\mathcal{E}_2$ are selfadjoint operator spaces, we show that $ρ_{\mathcal{E}_1}(X) = ρ_{\mathcal{E}_2}(X)$ 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 $ρ_{\mathcal{E}}(A)$ 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. |
| title | On the spectral radius of operator tuples |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2605.09354 |