Abstract and Applied Analysis
Volume 2011 (2011), Article ID 686832, 11 pages
http://dx.doi.org/10.1155/2011/686832
Research Article

Supercyclicity and Hypercyclicity of an Isometry Plus a Nilpotent

S. Yarmahmoodi,1 K. Hedayatian,2 and B. Yousefi3

1Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran 1477893855, Iran
2Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran
3Department of Mathematics, Payam-e-noor University, Shiraz 71955-1368, Iran

Received 30 December 2010; Accepted 28 March 2011

Academic Editor: John Rassias

Copyright © 2011 S. Yarmahmoodi et al. 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

Suppose that 𝑋 is a separable normed space and the operators 𝐴 and 𝑄 are bounded on 𝑋 . In this paper, it is shown that if 𝐴 𝑄 = 𝑄 𝐴 , 𝐴 is an isometry, and 𝑄 is a nilpotent then the operator 𝐴 + 𝑄 is neither supercyclic nor weakly hypercyclic. Moreover, if the underlying space is a Hilbert space and 𝐴 is a co-isometric operator, then we give sufficient conditions under which the operator 𝐴 + 𝑄 satisfies the supercyclicity criterion.