Operators characterized by certain Cauchy-Schwarz type inequalities
type: Item
instance of: scholarly article
title: Operators characterized by certain Cauchy-Schwarz type inequalities
zbMATH Open document ID: 0815.47028
author: Hideharu Watanabe
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1994-09-27T00:00:00Z
review text: Summary: A Hilbert space operator \(T\) satisfying \[ \begin{alignedat}{4} \text{either}\quad &(**) &&| (T\xi, \eta)|^ 2 \leq(| T| \xi,\xi) (| T| \eta, \eta), &&\text{for all } \xi,\eta\in {\mathcal H},\\ \text{or}\quad &(*) &&|( T\xi, \xi)|\leq (| T| \xi,\xi), &&\text{for all } \xi\in {\mathcal H} \end{alignedat} \] is studied. The condition \((**)\) defines a slightly larger class than the hyponormality, and for compact operators \((**)\) is equivalent to the normality. The condition \((*)\) is characterized by using an operator whose numerical radius is less than 1, and among other things we show that \((*)\) and the normality are equivalent for matrices. Moreover, we show that \((*)\) and the normality are equivalent for trace class operators in the Appendix.
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 621959
zbMATH Keywords: Hilbert space operator
zbMATH Keywords: hyponormality
zbMATH Keywords: compact operators
zbMATH Keywords: numerical radius
zbMATH Keywords: trace class operators
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195166132
OpenAlex ID: W2030370379
cites work: On Hyponormal Operators
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cites work: Q3909512
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cites work: Hyponormal operators
DOI: 10.2977/PRIMS/1195166132
description: scientific article; zbMATH DE number 621959