Operators Characterized by Certain Cauchy-Schwarz Type Inequalities

Abstract

A Hilbert space operator T satisfying either
(**) |(T ξ, η)|2 ≤ (|T| ξ,ξ) (|T| η,η) for all ξ, η ∈ ℋ,
or (*) |(T ξ, ξ)| ≤ (|(T|ξ,ξ) for all ξ ∈ ℋ,
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 Appendix.