A Quaternionic Hansen--Pedersen Inequality and its Application to $S_r$-Transforms

M. Fashandi

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arXiv · 2609.32623

A Quaternionic Hansen--Pedersen Inequality and its Application to $S_r$-Transforms

M. Fashandi

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Abstract

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.

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M. Fashandi. 2026-09-26. A Quaternionic Hansen--Pedersen Inequality and its Application to $S_r$-Transforms. https://arxiv.org/abs/2609.32623

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