Operators characterized by certain Cauchy-Schwarz type inequalities
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.
scientific article; zbMATH DE number 621959