International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 975745, 20 pages
http://dx.doi.org/10.1155/2012/975745
Research Article

On Some Normality-Like Properties and Bishop's Property ( 𝛽 ) for a Class of Operators on Hilbert Spaces

Sid Ahmed Ould Ahmed Mahmoud

Mathematics Department, College of Science, Al Jouf University, Al Jouf 2014, Saudi Arabia

Received 11 December 2011; Accepted 19 February 2012

Academic Editor: Shigeru Kanemitsu

Copyright © 2012 Sid Ahmed Ould Ahmed Mahmoud. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove some further properties of the operator 𝑇 ∈ [ 𝑛 Q N ] ( 𝑛 -power quasinormal, defined in Sid Ahmed, 2011). In particular we show that the operator 𝑇 ∈ [ 𝑛 Q N ] satisfying the translation invariant property is normal and that the operator 𝑇 ∈ [ 𝑛 Q N ] is not supercyclic provided that it is not invertible. Also, we study some cases in which an operator 𝑇 ∈ [ 2 Q N ] is subscalar of order 𝑚 ; that is, it is similar to the restriction of a scalar operator of order 𝑚 to an invariant subspace.