Abstract

In this paper we study the properties of bounded linear operators namely $A_p-$class and $A^ast_p-$class operators that satisfy $T^ast Tleq(T^{ast p} T^p )^{1over p}$ and $TT^astleq(T^{ast p} T^p )^{1over p}$ respectively.  We use some known operator inequalities and we  show that if $Tinmathcal{B}(mathscr{H})$ is an $A_p-$class or an $A^ast_p-$class operator, then  $r(T)=parallel Tparallel.$