Abstract and Applied Analysis

Fine Spectra of Symmetric Toeplitz Operators

Muhammed Altun

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Abstract

The fine spectra of 2-banded and 3-banded infinite Toeplitz matrices were examined by several authors. The fine spectra of n-banded triangular Toeplitz matrices and tridiagonal symmetric matrices were computed in the following papers: Altun, “On the fine spectra of triangular toeplitz operators” (2011) and Altun, “Fine spectra of tridiagonal symmetric matrices” (2011). Here, we generalize those results to the ( 2 n + 1 )-banded symmetric Toeplitz matrix operators for arbitrary positive integer n .

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 932785, 14 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495788

Digital Object Identifier
doi:10.1155/2012/932785

Mathematical Reviews number (MathSciNet)
MR2955013

Zentralblatt MATH identifier
1252.47030

Citation

Altun, Muhammed. Fine Spectra of Symmetric Toeplitz Operators. Abstr. Appl. Anal. 2012 (2012), Article ID 932785, 14 pages. doi:10.1155/2012/932785. https://projecteuclid.org/euclid.aaa/1355495788


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References

  • 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.
  • B. E. Rhoades, “The fine spectra for weighted mean operators,” Pacific Journal of Mathematics, vol. 104, no. 1, pp. 219–230, 1983.
  • 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.
  • 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.
  • J. T. Okutoyi, “On the spectrum of ${C}_{1}$ as an operator on bv,” University of Ankara. Faculty of Sciences. Communications, vol. 41, no. 1-2, pp. 197–207, 1992.
  • B. E. Rhoades and M. Yildirim, “Spectra and fine spectra for factorable matrices,” Integral Equations and Operator Theory, vol. 53, no. 1, pp. 127–144, 2005.
  • A. M. Akhmedov and F. Başar, “On spectrum of the Cesaro operator,” Proceedings of Institute of Mathematics and Mechanics. National Academy of Sciences of Azerbaijan, vol. 19, pp. 3–8, 2004.
  • 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.
  • M. Altun and V. Karakaya, “Fine spectra of lacunary matrices,” Communications in Mathematical Analysis, vol. 7, no. 1, pp. 1–10, 2009.
  • 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.
  • M. Altun, “On the fine spectra of triangular Toeplitz operators,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 8044–8051, 2011.
  • 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.
  • 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 (English Series), 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.
  • A. M. Akhmedov and S. R. El-Shabrawy, “On the fine spectrum of the operator ${\Delta }_{a,b}$ over the sequence space $c$,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 2994–3002, 2011.
  • P. D. Srivastava and S. Kumar, “Fine spectrum of the generalized difference operator ${\Delta }_{v}$ on sequence space ${\ell }_{1}$,” Thailand Journal of Mathematics, vol. 8, no. 2, pp. 7–19, 2010.
  • P. D. Srivastava and S. Kumar, “Fine spectrum of the generalized difference operator ${\Delta }_{uv}$ on sequence space ${\ell }_{1}$,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6407–6414, 2012.
  • E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, NY, USA, 1978.
  • S. Goldberg, Unbounded Linear Operators, Dover, New York, NY, USA, 1985.
  • 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.
  • 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, no. 18, pp. 3005–3013, 2005.
  • A. Wilansky, Summability through Functional Analysis, vol. 85 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1984.
  • R. P. Flowe and G. A. Harris, “A note on generalized Vandermonde determinants,” SIAM Journal on Matrix Analysis and Applications, vol. 14, no. 4, pp. 1146–1151, 1993.
  • F. L. Qian, “Generalized Vandermonde determinants,” Journal of Sichuan University. Natural Science Edition, vol. 28, no. 1, pp. 36–40, 1991.
  • Y.-M. Chen and H.-C. Li, “Inductive proofs on the determinants of generalized Vandermonde matrices,” International Journal of Computational and Applied Mathematics, vol. 5, no. 1, pp. 23–40, 2010.
  • J. P. Cartlidge, Weighted mean matrices as operators on $\ell ^{p}$ [Ph.D. thesis], Indiana University, Ind, USA, 1978.
  • M. Altun, “Fine spectra of tridiagonal symmetric matrices,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 161209, 10 pages, 2011.