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Let T be a bounded linear operator on a complex Hilbert space . For positive integers n and k, an operator T is called (n,k)-quasiparanormal if $\left|\right|T^{1+n}\left(T^{k}x\right){\left|\right|}^{1/(1+n)}\left|\right|T^k{x\left|\right|}^{n/(1+n)}\ge \left|\right|T\left(T^{k}x\right)\left|\right|$ for x ∈ . 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 (β); (4) quasinilpotent...