Ali Karaisa, Feyzi Başar

7294

On the fine spectrum of the generalized difference operator defined by a double sequential band matrix over the sequence space $ell_p$, (1 < p < ∞)

On the fine spectrum of the generalized difference operator defined by a double sequential band matrix over the sequence space $ell_p$, (1 < p < ∞)

The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg’s classification of the operator $B(tilde{r},tilde{s})$ defined by a double sequential band matrix over the sequence space $ell_p$ where 1 < p < ∞. These results are more general than the spectrum of the generalized difference operator B(r, s) over  $ell_p$ of Bilgiç and Furkan [Nonlinear Anal. 68(3)(2008), 499–506]. 
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  • [1] A.M. Akhmedov, F. Başar, On the fine spectrum of the Cesàro operator in c0, Math. J. Ibaraki Univ. 36 (2004), 25–32.
  • [2] A.M. Akhmedov, F. Başar, On the spectra of the difference operator ∆ over the sequence space `p, Demonstratio Math. 39 (3) (2006), 585–595.
  • [3] A.M. Akhmedov, F. Başar, On the fine spectra of the difference operator ∆ over the sequence space bvp, (1 ≤ p < ∞), Acta Math. Sin. Eng. Ser. 23 (10) (2007), 1757–1768.
  • [4] A.M. Akhmedov, S.R. El-Shabrawy, On the fine spectrum of the operator ∆a,b over the sequence space c, Comput. Math. Appl. 61 (10) (2011), 2994–3002.
  • [5] B. Altay, F. Başar, On the fine spectrum of the difference operator ∆ on c0 and c, Inform. Sci. 168 (2004), 217–224.
  • [6] B. Altay, F. Başar, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c0 and c, Int. J. Math. Math. Sci. 2005: 18 (2005), 3005–3013.
  • [7] B. Altay, M. Karakuş, On the spectrum and fine spectrum of the Zweier matrix as an operator on some sequence spaces, Thai J. Math. 3 (2005), 153–162.
  • [8] M. Altun, On the fine spectra of triangular Toeplitz operators, Appl. Math. Comput. 217 (20) (2011), 8044–8051.
  • [9] J. Appell, E. Pascale, A. Vignoli, Nonlinear Spectral Theory, de Gruyter Series in Nonlinear Analysis and Applications 10, Walter de Gruyter · Berlin · New York, 2004.
  • [10] F. Başar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55 (1) (2003), 136–147.
  • [11] H. Bilgiç, H. Furkan, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces `p and bvp (1 < p < ∞), Nonlinear Anal. 68 (3) (2008), 499–506.
  • [12] H. Bilgiç, H. Furkan, On the fine spectrum of the operator B(r, s, t) over the sequence spaces `1 and bv, Math. Comput. Modelling 45 (2007), 883–891.
  • [13] A. Karaisa, F. Başar, Fine spectra of upper triangular triple-band matrices over the sequence space `p, (0 < p < ∞), Abstr. Appl. Anal. 2013 Article ID 342682, 10 pages, 2013. doi:10.1155/2013/342682.
  • [14] A. Karaisa, F. Başar, Fine spectra of upper triangular triple-band matrix over the sequence space `p where (0 < p < 1), AIP Conference Proceedings 1470 (2012), 134–137.
  • [15] P.J. Cartlidge, Weighted mean matrices as operators on ` p, Ph.D. Dissertation, Indiana University, 1978.
  • [16] C. Coşkun, The spectra and fine spectra for p-Cesàro operators, Turkish J. Math. 21 (1997), 207–212.
  • [17] B. Choudhary, S. Nanda, Functional Analysis with Applications, John Wiley & Sons Inc. New York · Chichester · Brisbane · Toronto · Singapore, 1989.
  • [18] E. Dündar, F. Başar, On the fine spectrum of the upper triangle double band matrix ∆+ on the sequence space c0, Math. Commun. 18 (2013), 337–348.
  • [19] A. Farés, B. de Malafosse, Spectra of the operator of the first difference in sα, s 0α, s(c)α and lp(α) (1 ≤ p < ∞) and application to matrix transformations, Demonstratio Math. 41 (3) (2008), 661–676.
  • [20] H. Furkan, H. Bilgiç, B. Altay, On the fine spectrum of the operator B(r, s, t) over c0 and c, Comput. Math. Appl. 53 (2007), 989–998.
  • [21] H. Furkan, H. Bilgiç, F. Başar, On the fine spectrum of the operator B(r, s, t) over the sequence spaces `p and bvp, (1 < p < ∞), Comput. Math. Appl. 60 (7) (2010), 2141–2152.
  • [22] S. Goldberg, Unbounded Linear Operators, Dover Publications, Inc. New York, 1985.
  • [23] M. Gonzàlez, The fine spectrum of the Cesàro operator in `p (1 < p < ∞), Arch. Math. 44 (1985), 355–358.
  • [24] M. Imaninezhad, M.R. Miri, The dual space of the sequence space bvp, (1 ≤ p < ∞), Acta Math. Univ. Comen. 79 (1) (2010), 143–149.
  • [25] A. Karaisa, Spectrum and fine spectrum generalized diffrence over the sequence space `1, Math. Sci. Lett. 3 (3) (2014), 215–221.
  • [26] A. Karaisa, Fine spectra of upper triangular double-band matrices over the sequence space `p, (1 < p < ∞), Discrete Dyn. Nature Soc. 2012 Article ID 381069 (2012).
  • [27] F. Başar, A. Karaisa, Spectrum and fine spectrum of the upper triangular triple-band matrix over some sequence spaces, AIP Conference Proceedings 1676 (2015), 020085: 1-10, doi.org/10.1063/1.4930511.
  • [28] Saad R. El-Shabrawy, Spectra and fine spectra of certain lower triangular double-band matrices as operators on c0 J. Inequal. Appl. 2014 (2014), 1–9.
  • [29] K. Kayaduman, H. Furkan, The fine specta of the difference operator ∆ over the sequence spaces `1 and bv, Int. Math. Forum 1 (24) (2006), 1153–1160.
  • [30] E. Kreyszig,Introductory Functional Analysis with Applications, John Wiley & Sons Inc. New York · Chichester · Brisbane · Toronto, 1978.
  • [31] B. de Malafosse, Properties of some sets of sequences and application to the spaces of bounded difference sequences of order µ, Hokkaido Math. J. 31 (2002), 283–299.
  • [32] J.I. Okutoyi, On the spectrum of C1 as an operator on bv0, J. Austral. Math. Soc. Ser. A. 48 (1990), 79–86.
  • [33] J.I. Okutoyi, On the spectrum of C1 as an operator on bv, Commun. Fac. Sci. Univ. Ank. Ser. A1 41 (1992), 197–207.
  • [34] J.B. Reade, On the spectrum of the Cesaro operator, Bull. Lond. Math. Soc. 17 (1985), 263–267.
  • [35] B.E. Rhoades, The fine spectra for weighted mean operators, Pacific J. Math. 104 (1) (1983), 219–230.
  • [36] P.D. Srivastava, S. Kumar, Fine spectrum of the generalized difference operator ∆ν on sequence space `1, Thai J. Math. 8(2)(2010), 7–19.
  • [37] B.L. Panigrahi, P.D. Srivastava, Spectrum and fine spectrum of generalized second order difference operator ∆2 uv on sequence space c0, Thai J. Math. 9 (1) (2011), 57–74.
  • [38] P.D. Srivastava, S. Kumar, Fine spectrum of the generalized difference operator ∆uv on sequence space l1, Appl. Math. Comput. 218 (11) (2012), 6407–6414.
  • [39] R.B. Wenger, The fine spectra of Hölder summability operators, Indian J. Pure Appl. Math. 6 (1975), 695–712.
  • [40] M. Yeşilkayagil, F. Başar, On the fine spectrum of the operator defined by a lambda matrix over the sequence spaces of null and convergent sequences, Abstr. Appl. Anal. 2013, Article ID 687393, 13 pages, 2013. doi: 10.1155/2013/687393.
  • [41] M. Yıldırım, On the spectrum and fine spectrum of the compact Rhally operators, Indian J. Pure Appl. Math. 27 (8) (1996), 779–784.
  • [42] M. Yıldırım, On the spectrum of the Rhaly operators on c0 and c, Indian J. Pure Appl. Math. 29 (1998), 1301–1309.
  • [43] M. Yıldırım, On the spectrum of the Rhaly operators on `p, Indian J. Pure Appl. Math. 32 (2001), 191–198.
  • [44] M. Yıldırım, On the fine spectrum of the Rhaly operators on `p, East Asian Math. J. 20 (2004), 153–160.
Hacettepe Journal of Mathematics and Statistics-Cover
  • Yayın Aralığı: 6
  • Başlangıç: 2002
  • Yayıncı: Hacettepe Üniversitesi Fen Fakultesi
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