A Note on Class $Q(N)$ Operators

Abstract

Let $T$ be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. In this paper we introduce two new classes of operators: class $Q(N)$ and class $Q^*(N)$. An operator $T\in \mathcal{L}(\mathcal{H})$ is of class $Q(N)$ for a fixed real number $N\geq 1$, if $T$ satisfies $N\|Tx\|^{2} \leq \| T^2 x\|^{2}+ \| x\|^{2}$ for all $x\in \mathcal{H}$. And an operator $T\in \mathcal{L}(\mathcal{H})$ is of class $Q^*(N)$ for a fixed real number $N\geq 1$, if $T$ satisfies $N\|T^*x\|^{2} \leq \| T^2 x\|^{2}+ \| x\|^{2}$ for all $x\in \mathcal{H}$. We prove the basic properties of these classes of operators.