On (n, k)-quasiparanormal operators

Let T be a bounded linear operator on a complex Hilbert space H. For positive integers n and k, an operator T is called (n, k)-quasiparanormal if parallel to T1+n(T(k)x)parallel to(1/(1+n))parallel to T(k)x parallel to(n/(1+n)) >= parallel to T(T(k)x)parallel to for x is an element of H. The class of (n, k)-quasiparanormal operators contains the classes of n-paranormal and k-quasiparanormal operators. We consider some properties of (n, k)-quasiparanormal operators: (1) inclusion relations and examples; (2) a matrix representation and SVEP (single valued extension property); (3) ascent and Bishop's property (beta); (4) quasinilpotent part and Riesz idempotents for k-quasiparanormal operators.