Akhmedov, A. M.; El-Shabrawy, S. R. On the fine spectrum of the operator over the sequence space \(c\). (English) Zbl 1222.40002 Comput. Math. Appl. 61, No. 10, 2994-3002 (2011). Summary: We examine the fine spectrum of the generalized difference operator \(\Delta _{a,b}\) over the sequence space \(c\). The boundedness of the operator \(\Delta _{a,b}\) is proved. Also, the norm of this operator is found. The class of the operator \(\Delta _{a,b}\) includes some other special cases such as the generalized difference operator \(B(r,s)\) introduced by B. Altay and F. Başar [Int. J. Math. Math. Sci. 2005, No. 18, 3005–3013 (2005; Zbl 1098.39013)]. Our results not only generalize the corresponding results in the existing literature, but also give results for some more operators. Cited in 30 Documents MSC: 40C05 Matrix methods for summability 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 40H05 Functional analytic methods in summability 47A10 Spectrum, resolvent Keywords:spectrum of an operator; generalized difference operator; sequence space Citations:Zbl 1098.39013 PDFBibTeX XMLCite \textit{A. M. Akhmedov} and \textit{S. R. El-Shabrawy}, Comput. Math. Appl. 61, No. 10, 2994--3002 (2011; Zbl 1222.40002) Full Text: DOI References: [1] Kreyszig, E., Introductory Functional Analysis with Applications (1978), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York, Chichester, Brisbane, Toronto · Zbl 0368.46014 [2] Goldberg, S., Unbounded Linear Operators (1985), Dover Publications, Inc.: Dover Publications, Inc. New York [3] Akhmedov, A. M., On the spectrum of the generalized difference operator \(\Delta_\alpha\) over the sequence space \(l_p,(1 \leq p < \infty)\), Baku Univ. News J., Phys. Math. Sci. Ser., 3, 34-39 (2009) [4] Altay, B.; Başar, F., On the fine spectrum of the difference operator \(\Delta\) on \(c_0\) and \(c\), Inform. Sci., 168, 217-224 (2004) · Zbl 1085.47041 [5] Altay, B.; Başar, F., On the fine spectrum of the generalized difference operator \(B(r, s)\) over the sequence spaces \(c_0\) and \(c\), Int. J. Math. Math. Sci., 18, 3005-3013 (2005) · Zbl 1098.39013 [6] Altay, B.; Karakuş, M., On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces, Thai J. Math., 3, 2, 153-162 (2005) · Zbl 1183.47027 [7] Srivastava, P. D.; Kumar, S., On the fine spectrum of the generalized difference operator \(\Delta_v\) over the sequence space \(c_0\), Commun. Math. Anal., 6, 1, 8-21 (2009) · Zbl 1173.47022 [8] Wilansky, A., (Summability Through Functional Analysis. Summability Through Functional Analysis, North-Holland Mathematics Studies, vol. 85 (1984), North-Holland: North-Holland Amsterdam) · Zbl 0531.40008 [9] Akhmedov, A. M.; Başar, F., On the fine spectra of the difference operator \(\Delta\) over the sequence space \(l_p,(1 \leq p < \infty)\), Demonstratio Math., 39, 3, 585-595 (2006) · Zbl 1118.47303 [10] Akhmedov, A. M.; Başar, F., The fine spectra of the difference operator \(\Delta\) over the sequence space \(b v_p,(1 \leq p < \infty)\), Acta Math. Sin. (Engl. Ser.), 23, 10, 1757-1768 (2007) · Zbl 1134.47025 [11] Başar, F.; Altay, B., On the space of sequences of \(p\)-bounded variation and related matrix mappings, Ukrainian Math. J., 55, 1, 136-147 (2003) · Zbl 1040.46022 [12] De Malafosse, B., Properties of some sets of sequences and application to the spaces of bounded difference sequences of order \(\mu \), Hokkaido Math. J., 31, 283-299 (2002) · Zbl 1016.40002 [13] Bilgiç, H.; Furkan, H., On the fine spectrum of the generalized difference operator \(B(r, s)\) over the sequence spaces \(l_p\) and \(b v_p,(1 < p < \infty)\), Nonlinear Anal., 68, 499-506 (2008) · Zbl 1139.47005 [14] Srivastava, P. D.; Kumar, S., Fine spectrum of the generalized difference operator \(\Delta_v\) on sequence space \(l_1\), Thai J. Math., 8, 2, 221-233 (2010) · Zbl 1236.47029 [15] 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 J. Math., 9, 1, 57-74 (2011) · Zbl 1285.47040 [16] Furkan, H.; Bilgiç, H.; Başar, F., On the fine spectrum of the operator \(B(r, s, t)\) over the sequence spaces \(l_p\) and \(b v_p,(1 < p < \infty)\), Comput. Math. Appl., 60, 7, 2141-2152 (2010) · Zbl 1222.47050 [17] J.P. Cartlidge, Weighted mean matrices as operators on \(l^p\); J.P. Cartlidge, Weighted mean matrices as operators on \(l^p\) 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.