On $(n,k)$-quasiparanormal operators
Volume 209 / 2012
Studia Mathematica 209 (2012), 289-301
MSC: gxji@snnu.edu.cn
DOI: 10.4064/sm209-3-6
Abstract
Let $T$ be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. For positive integers $n$ and $k$, an operator $T$ is called $(n,k)$-quasiparanormal if \[ \|T^{1+n}(T^{k}x)\|^{{1}/{(1+n)}}\|T^{k}x\|^{{n}/{(1+n)}}\geq\|T(T^{k}x)\|\quad \hbox{for }x\in\mathcal{H}. \] 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.
