On operators satisfying T*m(T*|T|2kT)1/(k+1)Tm≥*m|T|2Tm

Let $T$ be a bounded linear operator acting on a complex Hilbert space $\hh$. In this paper we introduce the class, denoted $\qq(A(k)$, $m),$ of operators satisfying $T^{m*}(T^*|T|^{2k}T)^{1/(k+1)}T^m\geq T^{*m}|T|^2T^m$, where $m$ is a positive integer and $k$ is a positive real number and we prove basic structural properties of these operators. Using these results, we prove that if $P$ is the Riesz idempotent for isolated point $\lambda$ of the spectrum of $T\in \qq(A(k),m)$, then $P$ is self-adjoint, and we give a necessary and sufficient condition for $T\otimes S$ to be in $\qq(A(k),m)$ when $T$ and $S$ are both non-zero operators. Moreover, we characterize the quasinilpotent part $H_0(T-\lambda)$ of class $A(k)$ operator.