四元数 Hansen--Pedersen 不等式及其在 $S_r$-变换中的应用

中文摘要 AI 辅助

我们在右四元数 Hilbert 空间上，针对实算子凸函数建立了四元数 Hansen--Pedersen 不等式。证明基于自伴四元数算子的连续实函数演算、直和函数演算恒等式以及块酉构造。作为应用，设 $T = U|T|$ 为单射 $p$-亚正规算子。我们证明，对于 $0 < r \leq 1/2$，当 $0 < p \leq 1/2$ 时变换 $S_{r}(T) = U|T|^{r}U$ 是 $2p$-亚正规的，当 $1/2 < p \leq 1$ 时是亚正规的。我们还证明 $S_{r}(T)$ 对于 $r > 0$ 保持对数亚正规性。这些结果说明了四元数 Jensen 型算子不等式在研究与极分解相关的变换中的应用。

英文摘要

We establish a quaternionic Hansen--Pedersen inequality for real operator-convex functions on right quaternionic Hilbert spaces. The proof is based on the continuous real functional calculus for selfadjoint quaternionic operators, a direct-sum functional-calculus identity, and a block-unitary construction. As an application, let \(T = U|T|\) be an injective \(p\)-hyponormal operator. We prove that, for \(0 < r \leq 1/2\), the transform \(S_{r}(T) = U|T|^{r}U\) is \(2p\)-hyponormal when \(0 < p \leq 1/2\) and hyponormal when \(1/2 < p \leq 1\). We also show that \(S_{r}(T)\) preserves log-hyponormality for \(r > 0\). These results illustrate the use of quaternionic Jensen-type operator inequalities in the study of transforms associated with the polar decomposition.

发表机构

• Faculty of Mathematical Sciences, Ferdowsi University of Mashhad（法尔多斯马什哈德大学数学科学学院）

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