关于算子元组的谱半径
On the spectral radius of operator tuples
AI总结：
本文研究了算子空间结构对算子元组谱半径的影响，证明了谱半径仅依赖于底层规范空间，并探讨了不同结构下谱半径的差异及等价条件。
摘要：
在近期工作中，Shalit和Shamovich为每个在ℂ^d上的算子空间结构E定义了一个谱半径函数ρ_E。本文旨在阐明该谱半径如何依赖于算子空间结构。设V=(ℂ^d,‖·‖_V)为规范空间，且E为V的量化。我们证明，对于交换算子元组X，其谱半径仅依赖于底层规范空间；更准确地说，ρ_E(X)=max{‖λ‖_V:λ∈σ(X)}，其中σ(X)表示X的联合谱。相反，我们证明当dim V≥3时，对于某些矩阵元组X，ρ_{min(V)}(X)≠ρ_{max(V)}(X)。当E1和E2为自 adjoint 算子空间时，我们证明ρ_{E1}(X)=ρ_{E2}(X)对所有元组X成立当且仅当E1=E2。我们给出了该结果的两个证明；其中一个的关键成分是线性 pencil 关联于A的可逆域的特征化。最后，我们证明如果两个算子空间产生相同的谱半径函数，则对应的局部均匀有界NC函数代数在对应的NC单位球上一致。
英文摘要：
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.
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