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Chinese Journal of Mathematics
Volume 2013 (2013), Article ID 286748, 10 pages
http://dx.doi.org/10.1155/2013/286748
Research Article

Some Spectral Aspects of the Operator over the Sequence Spaces and

1Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar 751004, India
2Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar 751024, India

Received 22 July 2013; Accepted 28 August 2013

Academic Editors: Z. Huang, P. K. Sahoo, and W. Sun

Copyright © 2013 S. Dutta and P. Baliarsingh. 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

The main idea of the present paper is to compute the spectrum and the fine spectrum of the generalized difference operator over the sequence spaces . The operator denotes a triangular sequential band matrix defined by with for , where or , ; the set nonnegative integers and is either a constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. Finally, we obtain the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of the operator over the sequence spaces and . These results are more general and comprehensive than the spectrum of the difference operators , , , , and and include some other special cases such as the spectrum of the operators , , and over the sequence spaces or .