Spectra of the Lower Triangular Matrix $\mathbb{B}(r_1,\ldots,r_l;s_1,\ldots,s_{l'})$ over $c_0$
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Authors: S. K. MAHTO, A. PATRA AND P. D. SRIVASTAVA
DOI: 10.46793/KgJMat2203.369M
Abstract:
The infinite lower triangular matrix ????(r1,…,rl; s1,…,sl′) is considered over the sequence space c0, where l and l′ are positive integers. The diagonal and sub-diagonal entries of the matrix consist of the oscillatory sequences r = (rk(mod l)+1) and s = (sk(mod l′)+1), respectively. The rest of the entries of the matrix are zero. It is shown that the matrix represents a bounded linear operator. Then the spectrum of the matrix is evaluated and partitioned into its fine structures: point spectrum, continuous spectrum, residual spectrum, etc. In particular, the spectra of the matrix ????(r1,…,r4; s1,…,s6) are determined. Finally, an example is taken in support of the results.
Keywords:
Fine spectra, sequence space, lower triangular infinite matrix, point spectrum, continuous spectrum, residual spectrum.
References:
[1] A. M. Akhmedov and F. Başar, The fine spectra of the difference operator ▵ over the sequence space bvp(1 ≤ p < ∞), Acta Math. Sin. (Engl. Ser.) 23 (2007), 1757–1768.
[2] A. M. Akhmedov and S. R. El-Shabrawy, Spectra and fine spectra of lower triangular double-band matrices as operators on ℓp (1 ≤ p < ∞), Math. Slovaca 65 (2015), 1137–1152.
[3] B. Altay and F. Başar, On the fine spectrum of the difference operator ▵ on c0 and c, Inform. Sci. 168 (2004), 217–224.
[4] B. Altay and 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 (2005), 3005–3013.
[5] B. Altay and M. Karakuş, On the spectrum and the fine spectrum of the zweier matrix as an operator on some sequence spaces, Thai J. Math. 3 (2005), 153–162.
[6] H. Bilgiç and H. Furkan, On the fine spectrum of the generalized difference operator B(r,s) over the sequence spaces ℓp and bvp (1 < p < ∞), Nonlinear Anal. 68 (2008), 499–506.
[7] R. Birbonshi and P. D. Srivastava, On some study of the fine spectra of n-th band triangular matrices, Complex Anal. Oper. Theory 11 (2017), 739–753.
[8] E. Dündar and F. Başar, On the fine spectrum of the upper triangle double band matrix ▵ + on the sequence space c0, Math. Commun. 18 (2013), 337–348.
[9] S. Goldberg, Unbounded linear operators: Theory and applications, Courier Corporation, Mineola, New York, 2006.
[10] M. Gonzalez, The fine spectrum of the Cesàro operator in ℓp (1 < p < ∞), Arch. Math. (Basel) 44 (1985), 355–358.
[11] E. Kreyszig, Introductory functional analysis with applications, Wiley, New York, 1989.
[12] H. S. Özarslan and E. Yavuz, A new note on absolute matrix summability, J. Inequal. Appl. 2013 (2013), Paper ID 474, 7 pages.
[13] A. Patra, R. Birbonshi and P. D. Srivastava, On some study of the fine spectra of triangular band matrices, Complex Anal. Oper. Theory 13 (2019), 615–635.
[14] J. B. Reade, On the spectrum of the Cesàro operator, Bull. Lond. Math. Soc. 17 (1985), 263–267.
[15] B. E. Rhoades, The fine spectra for weighted mean operators, Pacific J. Math. 104 (1983), 219–230.
[16] B. E. Rhoades, The fine spectra for weighted mean operators in B(ℓp), Integral Equations Operator Theory 12 (1989), 82–98.
[17] P. D. Srivastava, Spectrum and fine spectrum of the generelized difference operators acting on some Banach sequence spaces - A review, Ganita 67 (2017), 57–68.
[18] P. D. Srivastava and S. Kumar, Fine spectrum of the generalized difference operator ▵ v on sequence space ℓ1, Thai J. Math. 8 (2010), 221–233.
[19] P. D. Srivastava and S. Kumar, Fine spectrum of the generalized difference operator ▵ uv on sequence space ℓ1, Appl. Math. Comput. 218 (2012), 6407–6414.
[20] M. Stieglitz and H. Tietz, Matrixtransformationen von folgenräumen eine ergebnisübersicht, Math. Z. 154 (1977), 1–16.
[21] A. E. Taylor and C. J. A. Halberg Jr, General theorems about a bounded linear operator and its conjugate, J. Reine Angew. Math. 198 (1957), 93–111.
[22] B. C. Tripathy and P. Saikia, On the spectrum of the Cesàro operator C1 on bv0 ∩ ℓ∞, Math. Slovaca 63 (2013), 563–572.
[23] R. B. Wenger, The fine spectra of the Hölder summability operators, Indian J. Pure Appl. Math. 6 (1975), 695–712.
[24] M. Yeşilkayagil and F. Başar, On the fine spectrum of the operator defined by the lambda matrix over the spaces of null and convergent sequences, Abstr. Appl. Anal. 2013 (2013), Article ID 687393, 13 pages.
[25] M. Yeşilkayagil and F. Başar, A survey for the spectrum of triangles over sequence spaces, Numer. Funct. Anal. Optim. (2019) (accepted).
[26] M. Yildirim, On the spectrum and fine spectrum of the compact Rhaly operators, Indian J. Pure Appl. Math. 27 (1996), 779–784.
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