On $(n,k)$-quasi class $Q$ Operators

Let $T$ be a bounded linear operator on a complex Hilbert space $H$. In this paper we introduce a new class of operators: $(n,k)$-quasi class $Q$ operators, superclass of $(n,k)$-quasi paranormal operators. An operator $T$ is said to be $(n,k)$-quasi class $Q$ if it satisfies $$\| T(T^{k}x)\|^{2} \leq \frac{1}{n+1}\left(\| T^{1+n}(T^{k}x)\|^{2} +n\| T^{k}x\|^{2}\right),$$ for all $x\in H$ and for some nonnegative integers $n$ and $k$. We prove the basic structural properties of this class of operators. It will be proved that If $T$ has a no non-trivial invariant subspace, then the nonnegative operator $$D=T^{*k}\left( T^{*(1+n)}T^{(1+n)}-\frac{n+1}{n}T^{*}T+I\right)T^{k}$$ is a strongly stable contraction. In section 4, we give some examples which compare our class with other known classes of operators and as a consequence we prove that $(n,k)$-quasi class $Q$ does not have SVEP property. In the last section we also characterize the $(n,k)$-quasi class $Q$ composition operators on Fock spaces.