On $(n,k)$-quasi-*-paranormal operators

For nonnegative integers $n$ and $k$, we introduce in this paper a new class of $(n,k)$-quasi-*-paranormal operators satisfying $$||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \geq ||T^*(T^{k}x)|| \makebox{\ for all} x \in H.$$ This class includes the class of $n$-*-paranormal operators and the class of $(1,k)$-quasi-*-paranormal operators contains the class of $k$-quasi-*-class $A$ operators. We study basic properties of $(n,k)$-quasi-*-paranormal operators: (1) inclusion relations and examples; (2) a matrix representation; (3) joint (approximate) point spectrum and single valued extension property.

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