On p-hyponormal operators on quaternionic Hilbert spaces

TL;DR

This work extends the theory of $p$-hyponormal operators to bounded right linear quaternionic operators on right quaternionic Hilbert spaces, leveraging the spherical spectrum, functional calculus, and polar decomposition. It establishes a quaternionic Furuta inequality for positive operators and analyzes the Aluthge transform in this setting, proving that the transform of a $p$-hyponormal operator with $p\in[\tfrac12,1)$ is hyponormal. The paper also develops a quaternionic generalized Cauchy–Schwarz framework (GCSI), showing that $p$-hyponormal operators satisfy GCSI and that GCSI operators are paranormal, thereby linking a broad class of operators to paranormality. Collectively, these results generalize key complex-analytic operator inequalities to quaternionic Hilbert spaces and provide tools for studying transforms and inequalities of quaternionic operators with potential applications in quaternionic quantum systems.

Abstract

This paper extends the notion of a p-hyponormal operator for a bounded right linear quaternionic operator defined on a right quaternionic Hilbert space. Several fundamental properties of complex p-hyponormal operators are investigated for the quaternionic ones. To develop the results, we prove the well-known Furuta inequality for quaternionic positive operators. This inequality opens the way to discuss the p-hyponormality of a quaternionic operator and its Aluthge transform. Finally, a new class of quaternionic operators is established between quaternionic p-hyponormal and quaternionic paranormal operators.

Theorems & Definitions (31)

• Proposition 1
• Definition 2
• Theorem 3
• Remark 4
• Theorem 5
• Lemma 6
• proof
• Corollary 7
• proof
• Theorem 8