A generalized H\xf6lder-type inequalities for measurable operators

Yazhou Han,Jingjing Shao

doi:10.1186/s13660-020-02467-w

Abstract

We prove a generalized Hölder-type inequality for measurable operators associated with a semi-finite von Neumann algebra which is a generalization of the result shown by Bekjan (Positivity 21:113–126, 2017). This also provides a generalization of the unitarily invariant norm inequalities for matrix due to Bhatia–Kittaneh, Horn–Mathisa, Horn–Zhan and Zou under a cohyponormal condition.

Highlights

Let Mn be the space of n × n complex matrices
A norm ||| · ||| on Mn is called unitarily invariant if |||UAV ||| = |||A||| for all A ∈ Mn and all unitary matrices U, V ∈ Mn
Let A, B ∈ Mn and r > 0, Horn and Mathisa proved in [15] the following Cauchy–Schwarz inequality for unitarily invariant norms

Summary

Introduction

Let Mn be the space of n × n complex matrices. A norm ||| · ||| on Mn is called unitarily invariant if |||UAV ||| = |||A||| for all A ∈ Mn and all unitary matrices U, V ∈ Mn. For A, B, X ∈ Mn. On the other hand, let A, B ∈ Mn and r > 0, Horn and Mathisa proved in [15] the following Cauchy–Schwarz inequality for unitarily invariant norms By the concept of uniform Hardy–Littlewood majorization Bekjan [8] gave a Höldertype inequality (1.4) for τ -measurable operators associated with a semi-finite von Neumann algebra and for symmetric Banach spaces norm. Let L0 be the set of all Lebesgue measurable functions on (0, ∞).

Results

Conclusion