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Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 381069, 19 pages

http://dx.doi.org/10.1155/2012/381069

Research Article

## Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space ,

Department of Mathematics, Faculty of Sciences, Konya Necmettin Erbakan University, Karacian Mahallesi, Ankara Caddesi 74, 42060 Konya, Turkey

Received 8 May 2012; Accepted 9 July 2012

Academic Editor: Antonia Vecchio

Copyright © 2012 Ali Karaisa. 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.

#### Linked References

- E. Kreyszig,
*Introductory Functional Analysis with Applications*, John Wiley & Sons, New York, NY, USA, 1978. - J. Appell, E. Pascale, and A. Vignoli,
*Nonlinear Spectral Theory*, vol. 10 of*de Gruyter Series in Nonlinear Analysis and Applications*, Walter de Gruyter, Berlin, Germany, 2004. View at Publisher · View at Google Scholar - S. Goldberg,
*Unbounded Linear Operators*, Dover Publications, New York, NY, USA, 1985. - R. B. Wenger, “The fine spectra of the Hölder summability operators,”
*Indian Journal of Pure and Applied Mathematics*, vol. 6, no. 6, pp. 695–712, 1975. View at Zentralblatt MATH - B. E. Rhoades, “The fine spectra for weighted mean operators,”
*Pacific Journal of Mathematics*, vol. 104, no. 1, pp. 219–230, 1983. View at Zentralblatt MATH - J. B. Reade, “On the spectrum of the Cesàro operator,”
*The Bulletin of the London Mathematical Society*, vol. 17, no. 3, pp. 263–267, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. M. Akhmedov and F. Başar, “On the fine spectrum of the Cesàro operator in ${c}_{0}$,”
*Mathematical Journal of Ibaraki University*, vol. 36, pp. 25–32, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. I. Okutoyi, “On the spectrum of ${C}_{1}$ as an operator on bv,”
*Communications de la Faculté des Sciences de l'Université d'Ankara A1*, vol. 41, no. 1-2, pp. 197–207, 1992. View at Zentralblatt MATH - M. Yıldırım, “On the spectrum and fine spectrum of the compact Rhaly operators,”
*Indian Journal of Pure and Applied Mathematics*, vol. 27, no. 8, pp. 779–784, 1996. View at Zentralblatt MATH - C. Coşkun, “The spectra and fine spectra for $p$-Cesàro operators,”
*Turkish Journal of Mathematics*, vol. 21, no. 2, pp. 207–212, 1997. - B. de Malafosse, “Properties of some sets of sequences and application to the spaces of bounded difference sequences of order $\mu $,”
*Hokkaido Mathematical Journal*, vol. 31, no. 2, pp. 283–299, 2002. View at Zentralblatt MATH - B. Altay and F. Başar, “On the fine spectrum of the difference operator $\mathrm{\Delta}$ on ${c}_{0}$ and $c$,”
*Information Sciences*, vol. 168, no. 1–4, pp. 217–224, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Bilgiç and H. Furkan, “On the fine spectrum of the operator $B(r,s,t)$ over the sequence spaces ${l}_{1}$ and $bv$,”
*Mathematical and Computer Modelling*, vol. 45, no. 7-8, pp. 883–891, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Furkan, H. Bilgiç, and F. Başar, “On the fine spectrum of the operator $B(r,s,t)$ over the sequence spaces ${\ell}_{p}$ and $b{v}_{p}$, $(1<p<\infty )$,”
*Computers & Mathematics with Applications*, vol. 60, no. 7, pp. 2141–2152, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. M. Akhmedov and S. R. El-Shabrawy, “On the fine spectrum of the operator ${\mathrm{\Delta}}_{a,b}$ over the sequence space $c$,”
*Computers & Mathematics with Applications*, vol. 61, no. 10, pp. 2994–3002, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. D. Srivastava and S. Kumar, “Fine spectrum of the generalized difference operator ${\mathrm{\Delta}}_{v}$ on sequence space ${l}_{1}$,”
*Thai Journal of Mathematics*, vol. 8, no. 2, pp. 7–19, 2010. View at Zentralblatt MATH - B. L. Panigrahi and P. D. Srivastava, “Spectrum and fine spectrum of generalized second order difference operator ${\mathrm{\Delta}}_{uv}^{2}$ on sequence space ${c}_{0}$,”
*Thai Journal of Mathematics*, vol. 9, no. 1, pp. 57–74, 2011. - P. D. Srivastava and S. Kumar, “Fine spectrum of the generalized difference operator ${\mathrm{\Delta}}_{uv}$ on sequence space ${l}_{1}$,”
*Applied Mathematics and Computation*, vol. 218, no. 11, pp. 6407–6414, 2012. View at Publisher · View at Google Scholar - M. González, “The fine spectrum of the Cesàro operator in ${\ell}_{p}$$(1<p<\infty )$,”
*Archiv der Mathematik*, vol. 44, no. 4, pp. 355–358, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. J. Cartlidge,
*Weighted mean matrices as operators on lp [Ph.D. thesis]*, Indiana University, 1978. - J. I. Okutoyi, “On the spectrum of ${C}_{1}$ as an operator on $b{v}_{0}$,”
*Journal of the Australian Mathematical Society A*, vol. 48, no. 1, pp. 79–86, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Yıldırım, “On the spectrum of the Rhaly operators on ${c}_{0}$ and $c$,”
*Indian Journal of Pure and Applied Mathematics*, vol. 29, no. 12, pp. 1301–1309, 1998. View at Zentralblatt MATH - M. Yıldırım, “On the spectrum of the Rhaly operators on ${l}_{p}$,”
*Indian Journal of Pure and Applied Mathematics*, vol. 32, no. 2, pp. 191–198, 2001. View at Zentralblatt MATH - M. Yıldırım, “The fine spectra of the Rhaly operators on ${c}_{0}$,”
*Turkish Journal of Mathematics*, vol. 26, no. 3, pp. 273–282, 2002. View at Zentralblatt MATH - M. Yıldırım, “On the spectrum of the Rhaly operators on
*bv*,”*East Asian Mathematical Journal*, vol. 18, pp. 21–41, 2002. - M. Yıldırım, “On the spectrum and fine spectrum of the compact Rhaly operators,”
*Indian Journal of Pure and Applied Mathematics*, vol. 34, no. 10, pp. 1443–1452, 2003. View at Zentralblatt MATH - M. Yildirim, “On the spectrum of the Rhaly operators on $b{v}_{0}$,”
*Communications of the Korean Mathematical Society*, vol. 18, no. 4, pp. 669–676, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Yildirim, “On the fine spectrum of the Rhaly operators on
*lp*,”*East Asian Mathematical Journal*, vol. 20, pp. 153–160, 2004. - B. Altay and F. Başar, “On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence spaces ${c}_{0}$ and $c$,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 18, pp. 3005–3013, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Kayaduman and H. Furkan, “The fine spectra of the difference operator $\mathrm{\Delta}$ over the sequence spaces ${l}_{1}$ and $bv$,”
*International Mathematical Forum*, vol. 1, no. 24, pp. 1153–1160, 2006. View at Zentralblatt MATH - A. M. Akhmedov and F. Başar, “On the fine spectra of the difference operator $\mathrm{\Delta}$ over the sequence space ${\ell}_{p}$,”
*Demonstratio Mathematica*, vol. 39, no. 3, pp. 585–595, 2006. View at Zentralblatt MATH - A. M. Akhmedov and F. Başar, “On the fine spectra of the difference operator $\mathrm{\Delta}$ over the sequence space $b{v}_{p}$, $(1\le p<\infty )$,”
*Acta Mathematica Sinica*, vol. 23, no. 10, pp. 1757–1768, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Başar and B. Altay, “On the space of sequences of $p$-bounded variation and related matrix mappings,”
*Ukrainian Mathematical Journal*, vol. 55, no. 1, pp. 136–147, 2003. View at Publisher · View at Google Scholar - H. Furkan, H. Bilgiç, and K. Kayaduman, “On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence spaces ${l}_{1}$ and $bv$,”
*Hokkaido Mathematical Journal*, vol. 35, no. 4, pp. 897–908, 2006. - H. Furkan, H. Bilgiç, and B. Altay, “On the fine spectrum of the operator $B(r,s,t)$ over ${c}_{0}$ and $c$,”
*Computers & Mathematics with Applications*, vol. 53, no. 6, pp. 989–998, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Altay and M. Karakuş, “On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces,”
*Thai Journal of Mathematics*, vol. 3, no. 2, pp. 153–162, 2005. View at Zentralblatt MATH - A. Farés and B. de Malafosse, “Spectra of the operator of the first difference in ${s}_{\alpha},{s}_{\alpha}^{0},{s}_{\alpha}^{(c)}$ and ${l}_{p}(\alpha )(1\le p<\infty )$
and application to matrix transformations,”
*Demonstratio Mathematica*, vol. 41, no. 3, pp. 661–676, 2008. View at Zentralblatt MATH - H. Bilgiç and H. Furkan, “On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence spaces ${l}_{p}$ and $b{v}_{p}$$(1<p<\infty )$,”
*Nonlinear Analysis*, vol. 68, no. 3, pp. 499–506, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Altun, “On the fine spectra of triangular Toeplitz operators,”
*Applied Mathematics and Computation*, vol. 217, no. 20, pp. 8044–8051, 2011. View at Publisher · View at Google Scholar - V. Karakaya and M. Altun, “Fine spectra of upper triangular double-band matrices,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 5, pp. 1387–1394, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Choudhary and S. Nanda,
*Functional Analysis with Applications*, John Wiley & Sons, New York, NY, USA, 1989. - S. R. El-Shabrawy, “On the fine spectrum of the generalized difference operator ${\mathrm{\Delta}}_{a,b}$ over the sequence ${l}_{p}$, $(1<p<\infty )$,”
*Applied Mathematics & Information Sciences*, vol. 6, no. 1, supplement, pp. 111S–118S, 2012.