×

On the fine spectrum of the operator defined by the lambda matrix over the spaces of null and convergent sequences. (English) Zbl 1315.47004

The paper attempts to determine the fine spectrum with respect to S. Goldberg’s classification of the operator defined by the lambda matrix over the sequence spaces \(c_0\) and \(c\) [Unbounded linear operators. Theory and applications. Reprint of the 1966 edition. New York, NY: Dover Publications, Inc. (1985; Zbl 0925.47001)]. It contains five sections. The first section is introductory in nature. The second section deals with subdivisions of the spectrum. In this section, the authors define the notions of point spectrum, continuous spectrum, residual spectrum, approximate point spectrum, defect spectrum, and compression spectrum. In the third section, the authors examine these different spectra for a matrix operator \(\Lambda\) on the sequence space \(c_0\). In the fourth section, the authors investigate the fine spectrum of the operator \(\Lambda\) over the sequence space \(c\). The authors also investigate some other relevant results within these sections. Finally, the authors summarize their major findings with an outlook to a forthcoming paper as concluding remarks in the fifth section.

MSC:

47A10 Spectrum, resolvent
40A05 Convergence and divergence of series and sequences
46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0925.47001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kreyszig, E., Introductory Functional Analysis with Applications, xiv+688 (1978), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0706.46001
[2] Appell, J.; De Pascale, E.; Vignoli, A., Nonlinear Spectral Theory. Nonlinear Spectral Theory, de Gruyter Series in Nonlinear Analysis and Applications, 10, xii+408 (2004), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 1056.47001 · doi:10.1515/9783110199260
[3] Goldberg, S., Unbounded Linear Operators, viii+199 (1985), New York, NY, USA: Dover Publications Inc., New York, NY, USA · Zbl 0925.47001
[4] Wenger, R. B., The fine spectra of the Hölder summability operators, Indian Journal of Pure and Applied Mathematics, 6, 6, 695-712 (1975) · Zbl 0362.40010
[5] Rhoades, B. E., The fine spectra for weighted mean operators, Pacific Journal of Mathematics, 104, 1, 219-230 (1983) · Zbl 0501.40007 · doi:10.2140/pjm.1983.104.219
[6] Reade, J. B., On the spectrum of the Cesàro operator, The Bulletin of the London Mathematical Society, 17, 3, 263-267 (1985) · Zbl 0548.47017 · doi:10.1112/blms/17.3.263
[7] Akhmedov, A. M.; Başar, F., On the fine spectrum of the Cesàro operator in \(c_0\), Mathematical Journal of Ibaraki University, 36, 25-32 (2004) · Zbl 1096.47031 · doi:10.5036/mjiu.36.25
[8] Okutoyi, J. T., On the spectrum of \(C_1\) as an operator on bv, Communications, Faculty of Science, University of Ankara. Series A1, 41, 1-2, 197-207 (1992) · Zbl 0831.47020
[9] Yıldırım, M., On the spectrum and fine spectrum of the compact Rhaly operators, Indian Journal of Pure and Applied Mathematics, 27, 8, 779-784 (1996) · Zbl 0859.47004
[10] Coşkun, C., The spectra and fine spectra for \(p\)-Cesàro operators, Turkish Journal of Mathematics, 21, 2, 207-212 (1997) · Zbl 0898.47021
[11] de Malafosse, B., Properties of some sets of sequences and application to the spaces of bounded difference sequences of order \(\mu \), Hokkaido Mathematical Journal, 31, 2, 283-299 (2002) · Zbl 1016.40002
[12] Altay, B.; Başar, F., On the fine spectrum of the difference operator \(\Delta\) on \(c_0\) and \(c\), Information Sciences, 168, 1-4, 217-224 (2004) · Zbl 1085.47041 · doi:10.1016/j.ins.2004.02.007
[13] Bilgiç, H.; Furkan, H., On the fine spectrum of the operator \(B(r, s, t)\) over the sequence spaces \(l_1\) and \(b v\), Mathematical and Computer Modelling, 45, 7-8, 883-891 (2007) · Zbl 1152.47024 · doi:10.1016/j.mcm.2006.08.008
[14] Furkan, H.; Bilgiç, H.; Başar, F., On the fine spectrum of the operator \(B(r, s, t)\) over the sequence spaces \(\ell_p\) and \(b v_p, (1 < p < ∞)\), Computers & Mathematics with Applications, 60, 7, 2141-2152 (2010) · Zbl 1222.47050 · doi:10.1016/j.camwa.2010.07.059
[15] Akhmedov, A. M.; El-Shabrawy, S. R., On the fine spectrum of the operator \(\Delta_{a, b}\) over the sequence space \(c\), Computers & Mathematics with Applications, 61, 10, 2994-3002 (2011) · Zbl 1222.40002 · doi:10.1016/j.camwa.2011.03.085
[16] Srivastava, P. D.; Kumar, S., Fine spectrum of the generalized difference operator \(\Delta_v\) on sequence space \(l_1\), Thai Journal of Mathematics, 8, 2, 221-233 (2010) · Zbl 1236.47029
[17] Panigrahi, B. L.; Srivastava, P. D., Spectrum and fine spectrum of generalized second order difference operator \(\Delta_{u v}^2\) on sequence space \(c_0\), Thai Journal of Mathematics, 9, 1, 57-74 (2011) · Zbl 1285.47040
[18] Srivastava, P. D.; Kumar, S., Fine spectrum of the generalized difference operator \(\Delta_{u v}\) on sequence space \(l_1\), Applied Mathematics and Computation, 218, 11, 6407-6414 (2012) · Zbl 1262.47055 · doi:10.1016/j.amc.2011.12.010
[19] Karaisa, A.; Başar, F., Fine spectra of upper triangular triple-band matrix over the sequence space \(\ell_p, (0 < p < ∞)\) · Zbl 1483.47061
[20] Mursaleen, M.; Noman, A. K., On the spaces of \(\lambda \)-convergent and bounded sequences, Thai Journal of Mathematics, 8, 2, 311-329 (2010) · Zbl 1218.46005
[21] Móricz, F., On \(\Lambda \)-strong convergence of numerical sequences and Fourier series, Acta Mathematica Hungarica, 54, 3-4, 319-327 (1989) · Zbl 0708.42004 · doi:10.1007/BF01952063
[22] Wilansky, A., Summability through Functional Analysis. Summability through Functional Analysis, North-Holland Mathematics Studies, 85, xii+318 (1984), Amsterdam, The Netherlands: North-Holland Publishing, Amsterdam, The Netherlands · Zbl 0531.40008
[23] González, M., The fine spectrum of the Cesàro operator in \(\ell_p(1 < p < ∞)\), Archiv der Mathematik, 44, 4, 355-358 (1985) · Zbl 0568.47021 · doi:10.1007/BF01235779
[24] Altay, B.; Karakuş, M., On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces, Thai Journal of Mathematics, 3, 2, 153-162 (2005) · Zbl 1183.47027
[25] Altun, M., On the fine spectra of triangular Toeplitz operators, Applied Mathematics and Computation, 217, 20, 8044-8051 (2011) · Zbl 1250.47033 · doi:10.1016/j.amc.2011.03.003
[26] Karakaya, V.; Altun, M., Fine spectra of upper triangular double-band matrices, Journal of Computational and Applied Mathematics, 234, 5, 1387-1394 (2010) · Zbl 1193.47006 · doi:10.1016/j.cam.2010.02.014
[27] Cass, F. P.; Rhoades, B. E., Mercerian theorems via spectral theory, Pacific Journal of Mathematics, 73, 1, 63-71 (1977) · Zbl 0341.40008 · doi:10.2140/pjm.1977.73.63
[28] Maddox, I. J., Elements of Functional Analysis, x+208 (1970), London, UK: Cambridge University Press, London, UK · Zbl 0193.08601
[29] Yeşilkayagil, M.; Başar, F., On the fine spectrum of the operator defined by a lambda matrix over the sequence space \(c_0\) and \(c\), Proceedings of the 1st International Conference on Analysis and Applied Mathematics (ICAAM ’12) · doi:10.1063/1.4747674
[30] Altay, B.; Başar, F., The fine spectrum and the matrix domain of the difference operator \(\Delta\) on the sequence space \(\ell_p, (0 < p < 1)\), Communications in Mathematical Analysis, 2, 2, 1-11 (2007) · Zbl 1173.47021
[31] Başar, F.; Altay, B., On the space of sequences of \(p\)-bounded variation and related matrix mappings, Ukrainian Mathematical Journal, 55, 1, 136-147 (2003) · Zbl 1040.46022 · doi:10.1023/A:1025080820961
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.