Shape, scale, and minimality of matrix ranges
Shape, scale, and minimality of matrix ranges
复制标题
矩阵范围的形状、尺度和极小值
DOI：
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发表时间：
2018
影响因子：
1.3
通讯作者：
Benjamin Passer
中科院分区：
文献类型：
--
作者：
Benjamin Passer
研究了算子元组$T$的矩阵值域$\mathcal{W}(T)$，它是闭有界矩阵凸集，它包含了在单位完全正映射到矩阵代数中的所有象集。$\mathcal{W}(T)$在多大程度上决定$T$？这个问题被考虑为最小元组，根据定义，它不允许具有与原始元组相同的矩阵范围的适当被和数。我们通过证明最小紧凑元组不需要由其矩阵范围唯一地确定来澄清文献中的结果。所构造的反例是这样的：矩阵值域$\mathcal{W}(T)$是位于其标量级$K$上的最小矩阵凸集，记为$\mathcal{W}^{\Text{min}}(K)$。另一方面，如果紧凑元组$T$满足$\mathcal{W}(T)=\mathcal{W}^{\Text{min}}(K)$，则即使$T$不正常，$K$的形状也是确定的。对于非紧元组，如果有足够多的孤立极值点，我们证明了类似的非唯一性结构也适用于$\mathcal{W}^{\Text{min}}(K)$。此外，我们还考虑了矩阵凸集的分次积的包含问题，得到了关于(非自伴)压缩到交换正规算子的扩张的一个潜在更严格的范数界。这一主张与一种独立的、明确的扩张程序相结合，具体改善了之前的界限。
We study the matrix range $\mathcal{W}(T)$ of an operator tuple $T$, which is the closed and bounded matrix convex set containing all images of $T$ under unital completely positive maps into matrix algebras. To what extent does $\mathcal{W}(T)$ determine $T$? This problem is considered for minimal tuples, which by definition admit no proper summand with the same matrix range as the original. We clarify results in the literature by showing that a minimal compact tuple need not be determined uniquely by its matrix range. The counterexamples constructed are such that the matrix range $\mathcal{W}(T)$ is the smallest matrix convex set lying over its scalar level $K$, denoted $\mathcal{W}^{\text{min}}(K)$. On the other hand, if a compact tuple $T$ satisfies $\mathcal{W}(T) = \mathcal{W}^{\text{min}}(K)$, then the shape of $K$ is determined, even if $T$ is not normal. For non-compact tuples, we show that similar non-uniqueness constructions apply to $\mathcal{W}^{\text{min}}(K)$ if there are sufficiently many isolated extreme points of $K$. In addition, we consider containment problems for graded products of matrix convex sets, leading to a potentially stricter norm bound on the dilation of (non self-adjoint) contractions to commuting normal operators. This claim is paired with an independent, explicit dilation procedure that concretely improves previous bounds.
影响因子：
0.8
作者：
T.-L. Kriel
通讯作者：
T.-L. Kriel