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Some Spectral Aspects of the Operator over the Sequence Spaces and

DOI: 10.1155/2013/286748

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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 . 1. Introduction, Preliminaries, and Definitions In analysis, operator theory is one of the important branch of mathematics which has vast applications in the field applied science and engineering. Operator theory deals with the study related to different properties of operators such as their inverse, spectrum, and fine spectrum. Since the spectrum of a bounded linear operator generalizes the notion of eigen values of the corresponding matrix, therefore, the study of spectrum of an operator takes a prominent position in solving many scientific and engineering problems. Hence, mathematicians and researchers have devoted their works in achieving new ideas and concepts in the concerned field. For instance, the fine spectrum of the Cesàro operator on the sequence space for has been studied by Gonzalez [1]. Okutoyi [2] computed the spectrum of the Cesàro operator over the sequence space . The fine spectra of the Cesàro operator over the sequence space have been determined by Akhmedov and Ba?ar [3]. Akhmedov and Ba?ar [4, 5] have studied the fine spectrum of the difference operator over the sequence spaces and , where . Altay and Ba?ar [6] have determined the fine spectrum of the difference operator over the sequence spaces , for . The fine spectrum of the difference operator over the sequence spaces and was investigated by Kayaduman and Furkan [7]. Srivastava and Kumar [8] have examined the fine spectrum of the generalized difference operator over the sequence space . Recently, the spectrum of the generalized difference operator over the sequence spaces and has been studied by Dutta and Baliarsingh [9, 10], respectively. The main focus of this paper is to define the difference operator and establish its spectral characterization with respect to the Goldberg’s classifications. Let be either constant or strictly

References

[1]  M. Gonzalez, “The fine spectrum of the Cesàro operator in ?p ,” Archiv der Mathematik, vol. 44, no. 4, pp. 355–358, 1985.
[2]  J. T. Okutoyi, “On the spectrum of as an operator on bv,” Communications de la Faculté des Sciences de l'Université d'Ankara A, vol. 141, pp. 197–207, 1992.
[3]  A. M. Akhmedov and F. Ba?ar, “The fine spectra of Cesàro operator over the sequence space ,” Mathematical Journal of Okayama University, vol. 50, pp. 135–147, 2008.
[4]  A. M. Akhmedov and F. Ba?ar, “The fine spectra of the difference operator Δ over the sequence space ?p, ,” Demonstratio Mathematica, vol. 39, no. 3, pp. 586–595, 2006.
[5]  A. M. Akhmedov and F. Ba?ar, “The fine spectra of the difference operator Δ over the sequence space , ,” Acta Mathematica Sinica (English Series), vol. 23, no. 10, pp. 1757–1768, 2007.
[6]  B. Altay and F. Ba?ar, “The fine spectrum and the matrix domain of the difference operator Δ on the sequence space ?p, ,” Communications in Mathematical Analysis, vol. 2, pp. 1–11, 2007.
[7]  K. Kayaduman and H. Furkan, “The fine spectra of the difference operator Δ over the sequence spaces ?1 and bv,” International Mathematical Forum, vol. 1, no. 24, pp. 1153–1160, 2006.
[8]  P. D. Srivastava and S. Kumar, “On the fine spectrum of the generalized difference operator over the sequence space c0,” Communications in Mathematical Analysis, vol. 6, no. 1, pp. 8–21, 2009.
[9]  S. Dutta and P. Baliarsingh, “On a spectral classification of the operator over the sequence space c0,” Proceedings of the National Academy of Sciences, India A. In press.
[10]  S. Dutta and P. Baliarsingh, “On the fine spectra of the generalized rth difference operator on the sequence space ?1,” Applied Mathematics and Computation, vol. 219, pp. 1776–1784, 2012.
[11]  B. Altay and F. Ba?ar, “On the fine spectrum of the generalized difference operator over the sequence spaces c0 and c,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 18, pp. 3005–3013, 2005.
[12]  B. Altay and F. Ba?ar, “On the fine spectrum of the difference operator Δ on c0 and c,” Information Sciences, vol. 168, no. 1–4, pp. 217–224, 2004.
[13]  A. M. Akhmedov and S. R. El-Shabrawy, “On the fine spectrum of the operator Δa,b over the sequence space c,” Computers and Mathematics with Applications, vol. 61, no. 10, pp. 2994–3002, 2011.
[14]  B. L. Panigrahi and P. D. Srivastava, “Spectrum and fine spectrum of generalized second order difference operator on sequence space c0,” Thai Journal of Mathematics, vol. 9, no. 1, pp. 57–74, 2011.
[15]  S. Dutta and P. Baliarsingh, “On the spectrum of 2-nd order generalized difference operator over the sequence space c0,” Boletim da Sociedade Paranaense de Matemática, vol. 31, no. 2, pp. 235–244, 2013.
[16]  H. Furkan, H. Bilgi?, and F. Ba?ar, “On the fine spectrum of the operator over the sequence spaces ?p and bvp, ,” Computers and Mathematics with Applications, vol. 60, no. 7, pp. 2141–2152, 2010.
[17]  E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, NY, USA, 1978.
[18]  S. Goldberg, Unbounded Linear Operators, Dover Publications, New York, NY, USA, 1985.
[19]  I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London, UK, 1970.
[20]  B. Choudhary and S. Nanda, Functional Analysis with Applications, John Wiley & Sons, New York, NY, USA, 1989.

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