Some Properties of M-Class Q and M-Class Q^* Operators

Let T be a bounded linear operator on a complex Hilbert space \mathcalH. In this paper we introduce a new class of operators: M-class Q^* operators. An operator T\in \mathcalL(\mathcalH) is of M-class Q^*, for a fixed real number M\geq 1, if T satisfies \|T^*x\|^2 \leq \frac12(M^2\| T^2x\|^2 +\|x\|^2), for all x\in\mathcalH. We prove the basic properties of this class of operators and the M-class Q operators.