On $(n,k)$-quasiparanormal operators

Jiangtao Yuan¹ ; Guoxing Ji²

¹College of Mathematics and Information Science Shaanxi Normal University Xian 710062, China and School of Mathematics and Information Science Henan Polytechnic University Jiaozuo 454000, Henan Province, China
²College of Mathematics and Information Science Shaanxi Normal University Xian 710062, China

Studia Mathematica, Tome 209 (2012) no. 3, pp. 289-301

Résumé

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.

Détail

DOI : 10.4064/sm209-3-6

Keywords: bounded linear operator complex hilbert space mathcal positive integers operator called quasiparanormal geq quad hbox mathcal class quasiparanormal operators contains classes n paranormal k quasiparanormal operators consider properties quasiparanormal operators inclusion relations examples nbsp matrix representation svep single valued extension property ascent bishops property beta quasinilpotent part riesz idempotents k quasiparanormal operators