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A Quaternionic Hansen--Pedersen Inequality and its Application to Sr-Transforms
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 ≤ 1/2, the transform Sr(T) = U|T|rU is 2p-hyponormal when 0 < p ≤ 1/2 and hyponormal when 1/2 < p ≤ 1. We also show that Sr(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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BibTeX
@misc{fashandi2026quaternionic,
title = {{A Quaternionic Hansen--Pedersen Inequality and its Application to $S_r$-Transforms}},
author = {M. Fashandi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32623},
eprint = {2609.32623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32623},
}
RIS
TY - UNPB
TI - A Quaternionic Hansen--Pedersen Inequality and its Application to S_r-Transforms
AU - M. Fashandi
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32623
UR - https://arxiv.org/abs/2609.32623
AB - 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 ≤ 1/2, the transform S_r(T) = U|T|^rU is 2p-hyponormal when 0 < p ≤ 1/2 and hyponormal when 1/2 < p ≤ 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.
ER -