Operators Characterized by Certain Cauchy-Schwarz Type Inequalities

Abstract

A Hilbert space operator $T$ satisfying

is studied. The condition ($\ast \ast$) defines a slightly larger class than the hyponormality, and for compact operators ($\ast \ast$) is equivalent to the normality. The condition ($\ast$) is characterized by using an operator whose numerical radius is less than 1, and among other things we show that ($\ast$) and the normality are equivalent for matrices. Moreover, we show that ($\ast$) and the normality are equivalent for trace class operators in Appendix.