Spectrum and fine spectrum of the upper triangular matrix U(r, s) over the sequence spaces
DOI:
https://doi.org/10.4067/S0716-09172015000200001Keywords:
Spectrum of an operator, Matrix mapping, Sequence space.Abstract
Fine spectra of various matrix operators on different sequence spaces have been investigated by several authors. Recently, some authors have determined the approximate point spectrum, the defect spectrum and the compression spectrum of various matrix operators on different sequence spaces. Here in this article we have determined the spectrum and fine spectrum of the upper triangular matrix U(r,s) on the sequence space cs. In a further development, we have also determined the approximate point spectrum, the defect spectrum and the compression spectrum of the operator U(r,s) on the sequence space cs.References
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[20] V. Karakaya and M. Altun, Fine spectra of upper triangular doubleband matrices, J. Comp. Appl. Math., 234, pp. 1387-1394, (2010).
[21] K. Kayaduman and H. Furkan, The fine spectra of the difference operator A over the sequence spaces c1 and bv ,Int. Math. Forum, 1 (24), pp. 1153-1160, (2006).
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[24] B. L. Panigrahi and P. D. Srivastava, Spectrum and the fine spectrum of the generalised second order difference operator AUv on sequence space c0, Thai J. Math., Vol. 9, No. 1, pp. 57-74, (2011).
[25] B. L. Panigrahi and P. D. Srivastava, Spectrum and fine spectrum of the generalized second order forward difference operator Afavw on the sequence space c1, Demonstratio Mathematica, Vol. XLV, No. 3, (2012).
[26] D. Rath and B. C. Tripathy, On the Banach algebra of triangular conservative matrices of operators, J. Math. Anal. Appl.; 197(3), pp. 743-751, (1994).
[27] B. E. Rhoades, The fine spectra for weighted mean operators, Pac. J. Math., 104 (1), pp. 219-230, (1983).
[28] P. D. Srivastava and S. Kumar, Fine spectrum of the generalized difference operator Av on the sequence space c1, Thai J. Math., 8 (2), pp. 221-233, (2010).
[29] B. C. Tripathy and A. Paul, Spectra of the operator B(f,g) on the vector valued sequence space c0(X), Bol. Soc. Parana. Mat., 31 (1), pp. 105-111, (2013).
[30] B. C. Tripathy and A. Paul, The Spectrum of the operator D(r, 0, 0, s) over the sequence space c0 and c; Kyungpook Math. J., 53 (2), pp. 247- 256, (2013).
[31] B. C. Tripathy and P. Saikia, On the spectrum of the Cesàro operator C1 bv n tm, Math. Slovaca, 63 (3), pp. 563-572, (2013).
[32] A. Wilansky, Summability Through Functional Analysis, North Holland, (1984).
[2] A. M. Akhmedov and F. Basar, The fine spectra of the Cesaro operator C1 over the sequence space bvp (1 < p < to), Math. J. Okayama Univ. 50, pp. 135-147, (2008).
[3] B. Altay and F. Basar, On the fine spectrum of the difference operator A on c0 and c, Inf. Sci., 168, pp. 217-224, (2004).
[4] B. Altay and F. Basar, 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, pp. 3005-3013, (2005).
[5] B. Altay and M. Karaku¸s, On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces, Thai J. Math., 3 (2), pp. 153-162, (2005).
[6] M. Altun, On the fine spectra of triangular Toeplitz operators, Appl. Math. Comput., 217, pp. 8044-8051, (2011).
[7] M. Altun, Fine spectra of tridiagonal symmetric matrices, Int. J. Math. Math. Sci., Article ID 161209, (2011).
[8] J. Appell, E. Pascale, A. Vignoli, Nonlinear Spectral Theory, Walter de Gruyter, Berlin, New York, (2004).
[9] F. Ba¸sar, N. Durna, M. Yildirim, Subdivisions of the spectra for generalized difference operator over certain sequence spaces, Thai J. Math., 9 (2), pp. 279-289, (2011).
[10] F. Ba¸sar, Summability Theory and Its Applications, Bentham Science Publishers,e-books, Monographs, Istanbul, (2012).
[11] H. Bilgi¸c and H.Furkan, On the fine spectrum of operator B(r, s, t) over the sequence spaces c1 and bv, Math Comput. Model., 45, pp. 883-891, (2007).
[12] H. Bilgi¸c and H.Furkan, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c1 and bvp (1 <p< to), Nonlinear Anal., 68, pp. 499-506, (2008).
[13] J. P. Cartlitdge, Weighted mean matrices as operators on cp, Ph. D. dissertation, Indiana University, Indiana, (1978).
[14] H. Furkan, H. Bilgi¸c and K. Kayaduman, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c1 and bv, Hokkaido Math. J. 35, pp. 893-904, (2006).
[15] H. Furkan, H. Bilgi¸c and B. Altay, On the fine spectrum of operator B(r, s, t) over c0 and c, Comput. Math. Appl., 53, pp. 989-998, (2007).
[16] H. Furkan, H. Bilgi¸c and F. Basar, On the fine spectrum of operator B(r, s, t) over the sequence spaces cp and bvp,(1 <p< to), Comput. Math. Appl., 60, pp. 2141-2152, (2010).
[17] S. Goldberg, Unbounded Linear Operators-Theory and Applications, Dover Publications, Inc, New York.
[18] M. Gonzalez, The fine spectrum of the Cesàro operator in £p, Arch. Math., 44, pp. 355-358, (1985).
[19] A. Karaisa and F. Basar, Fine spectrum of upper triangular tripleband matrices over the sequence space cp (0 <p< to, Abst. Appl. Anal., Article ID 342682, (2013).
[20] V. Karakaya and M. Altun, Fine spectra of upper triangular doubleband matrices, J. Comp. Appl. Math., 234, pp. 1387-1394, (2010).
[21] K. Kayaduman and H. Furkan, The fine spectra of the difference operator A over the sequence spaces c1 and bv ,Int. Math. Forum, 1 (24), pp. 1153-1160, (2006).
[22] E. Kreysziag, Introductory Functional Analysis with Application, John Wiley and Sons, (1989).
[23] J. I. Okutoyi, On the spectrum of C1 as an operator on bv0, J. Austral. Math. Soc. (Series A) 48, pp. 79-86, (1990).
[24] B. L. Panigrahi and P. D. Srivastava, Spectrum and the fine spectrum of the generalised second order difference operator AUv on sequence space c0, Thai J. Math., Vol. 9, No. 1, pp. 57-74, (2011).
[25] B. L. Panigrahi and P. D. Srivastava, Spectrum and fine spectrum of the generalized second order forward difference operator Afavw on the sequence space c1, Demonstratio Mathematica, Vol. XLV, No. 3, (2012).
[26] D. Rath and B. C. Tripathy, On the Banach algebra of triangular conservative matrices of operators, J. Math. Anal. Appl.; 197(3), pp. 743-751, (1994).
[27] B. E. Rhoades, The fine spectra for weighted mean operators, Pac. J. Math., 104 (1), pp. 219-230, (1983).
[28] P. D. Srivastava and S. Kumar, Fine spectrum of the generalized difference operator Av on the sequence space c1, Thai J. Math., 8 (2), pp. 221-233, (2010).
[29] B. C. Tripathy and A. Paul, Spectra of the operator B(f,g) on the vector valued sequence space c0(X), Bol. Soc. Parana. Mat., 31 (1), pp. 105-111, (2013).
[30] B. C. Tripathy and A. Paul, The Spectrum of the operator D(r, 0, 0, s) over the sequence space c0 and c; Kyungpook Math. J., 53 (2), pp. 247- 256, (2013).
[31] B. C. Tripathy and P. Saikia, On the spectrum of the Cesàro operator C1 bv n tm, Math. Slovaca, 63 (3), pp. 563-572, (2013).
[32] A. Wilansky, Summability Through Functional Analysis, North Holland, (1984).
How to Cite
[1]
B. C. Tripathy and R. Das, “Spectrum and fine spectrum of the upper triangular matrix U(r, s) over the sequence spaces”, Proyecciones (Antofagasta, On line), vol. 34, no. 2, pp. 107-125, 1.
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