c数值范围和酉膨胀

IF 0.5 Q3 MATHEMATICS ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2023-04-17 DOI:10.1007/s44146-023-00071-0

Chi-Kwong Li

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引用次数: 0

摘要

对于$n\times n$复矩阵C，作用于至少n维希尔伯特空间的有界线性算子T的C-数值范围是复数集$\textrm{tr}\,(CX\,^*\,TX)$，其中X是满足$X^*X = I_n$的部分等距。证明了$$\begin{aligned} \textbf{cl}(W_C(T)) = \cap \{\textbf{cl}(W_C(U)): U \hbox { is a unitary dilation of } T\} \end{aligned}$$对于任何收缩T当且仅当C是1阶正规矩阵。

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The C-numerical range and unitary dilations

For an $n\times n$ complex matrix C, the C-numerical range of a bounded linear operator T acting on a Hilbert space of dimension at least n is the set of complex numbers $\textrm{tr}\,(CX\,^*\,TX)$, where X is a partial isometry satisfying $X^*X = I_n$. It is shown that

$$\begin{aligned} \textbf{cl}(W_C(T)) = \cap \{\textbf{cl}(W_C(U)): U \hbox { is a unitary dilation of } T\} \end{aligned}$$

for any contraction T if and only if C is a rank one normal matrix.

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