Some Properties of (M,k)-quasi Paranormal Operators on Hilbert Spaces

Let H be a complex Hilbert space and let T represent a bounded linear operator on H. In this paper we introduce, a new class of non-normal operators, the (M,k)-quasi paranormal operator. An operator T is said to be (M,k)-quasi paranormal operator, for a non-negative integer k and a real positive number M, if it satisfies: ||Tk+1x||2 < M ||Tk+2x|| ||Tkx||, for every x in H. This new class of operators is a generalization of some of the non-normal operators, such as the k-quasi paranormal and M-paranormal operators. We prove the basic properties, the structural and spectral properties and also the matrix representation of this new class of operators.