Commun. Korean Math. Soc. 2017; 32(3): 661-676
Online first article March 28, 2017 Printed July 31, 2017
https://doi.org/10.4134/CKMS.c160191
Copyright © The Korean Mathematical Society.
Mohammad H. M. Rashid
Mu'tah University
Let $T$ be a bounded linear operator acting on a complex Hilbert space $\hh$. In this paper we introduce the class, denoted $\qq(A(k)$, $m),$ of operators satisfying $T^{m*}(T^*|T|^{2k}T)^{1/(k+1)}T^m\geq T^{*m}|T|^2T^m$, where $m$ is a positive integer and $k$ is a positive real number and we prove basic structural properties of these operators. Using these results, we prove that if $P$ is the Riesz idempotent for isolated point $\lambda$ of the spectrum of $T\in \qq(A(k),m)$, then $P$ is self-adjoint, and we give a necessary and sufficient condition for $T\otimes S$ to be in $\qq(A(k),m)$ when $T$ and $S$ are both non-zero operators. Moreover, we characterize the quasinilpotent part $H_0(T-\lambda)$ of class $A(k)$ operator.
Keywords: Riesz idempotent, tensor product, class $A(k)$, $m$-quasi-class $A(k)$
MSC numbers: 47A55, 47A10, 47A11
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