Abstract and Applied Analysis

Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space p ( 0 < p < )

Ali Karaisa and Feyzi Başar

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Abstract

The fine spectra of lower triangular triple-band matrices have been examined by several authors (e.g., Akhmedov (2006), Başar (2007), and Furken et al. (2010)). Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space p . The operator A ( r , s , t ) on sequence space on p is defined by A ( r , s , t ) x = ( r x k + s x k + 1 + t x k + 2 ) k = 0 , where x = ( x k ) p , with 0 < p < . In this paper we have obtained the results on the spectrum and point spectrum for the operator A ( r , s , t ) on the sequence space p . Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator A ( r , s , t ) on the sequence space p are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator A ( r , s , t ) over the space p and we give some applications.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 342682, 10 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511782

Digital Object Identifier
doi:10.1155/2013/342682

Mathematical Reviews number (MathSciNet)
MR3035358

Zentralblatt MATH identifier
06209258

Citation

Karaisa, Ali; Başar, Feyzi. Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ${\ell }_{p}$ ( $\mathrm{0}&lt;p&lt;\infty $ ). Abstr. Appl. Anal. 2013 (2013), Article ID 342682, 10 pages. doi:10.1155/2013/342682. https://projecteuclid.org/euclid.aaa/1393511782


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References

  • M. Gonzàlez, “The fine spectrum of the Cesàro operator in ${\ell }_{p}$ ($1<p<\infty $),” Archiv der Mathematik, vol. 44, no. 4, pp. 355–358, 1985.
  • P. J. Cartlidge, Weighted mean matrices as operators on $\ell ^{p}$ [Ph.D. dissertation], Indiana University, 1978.
  • J. I. Okutoyi, “On the spectrum of \emphC$_{1}$ as an operator on $b{v}_{0}$,” Australian Mathematical Society A, vol. 48, no. 1, pp. 79–86, 1990.
  • J. I. Okutoyi, “On the spectrum of \emphC$_{1}$ as an operator on bv,” Communications A, vol. 41, no. 1-2, pp. 197–207, 1992.
  • M. Y\ild\ir\im, “On the spectrum of the Rhaly operators on ${\ell }_{p}$,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 2, pp. 191–198, 2001.
  • B. Altay and F. Başar, “On the fine spectrum of the difference operator $\Delta $ on ${c}_{0}$ and $c$,” Information Sciences, vol. 168, no. 1–4, pp. 217–224, 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.
  • A. M. Akhmedov and F. Başar, “On the fine spectra of the difference operator $\Delta $ over the sequence space ${\ell }_{p}$, ($1\leq p<\infty $),” Demonstratio Mathematica, vol. 39, no. 3, pp. 585–595, 2006.
  • A. M. Akhmedov and F. Başar, “The fine spectra of the difference operator $\Delta $ over the sequence space $b{v}_{p}$, ($1\leq p<\infty $),” Acta Mathematica Sinica, vol. 23, no. 10, pp. 1757–1768, 2007.
  • 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.
  • 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.
  • 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.
  • A. Karaisa, “Fine spectra of upper triangular double-band matrices over the sequence space ${\ell }_{p}$, ($1<p<\infty $),” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 381069, 19 pages, 2012.
  • 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.
  • S. Goldberg, Unbounded Linear Operators, Dover Publications, New York, NY, USA, 1985.
  • B. Choudhary and S. Nanda, Functional Analysis with Applications, John Wiley & Sons, New York, NY, USA, 1989.