×

Fine spectra of upper triangular triple-band matrices over the sequence space \(\ell_p\) (\(0 < p < \infty\)). (English) Zbl 1483.47061

Summary: The fine spectra of lower triangular triple-band matrices have been examined by several authors, e.g., [A. M. Akhmedov and the second author, Demonstr. Math. 39, No. 3, 585–595 (2006; Zbl 1118.47303); Acta Math. Sin., Engl. Ser. 23, No. 10, 1757–1768 (2007; Zbl 1134.47025); H. Furkan et al., Comput. Math. Appl. 60, No. 7, 2141–2152 (2010; Zbl 1222.47050)]. Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space \(\ell_p\). The operator \(A(r, s, t)\) on sequence space on \(\ell_p\) is defined by \(A(r, s, t)x = (rx_k + sx_{k + 1} + tx_{k + 2})^\infty_{k = 0}\), where \(x = (x_k) \in \ell_p\), with \(0 < p < \infty\). In this paper we have obtained the results on the spectrum and point spectrum for the operator \(A(r, s, t)\) on the sequence space \(\ell_p\). Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator \(A(r, s, t)\) on the sequence space \(\ell_p\) are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator \(A(r, s, t)\) over the space \(\ell_p\) and we give some applications.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10 Spectrum, resolvent
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gonzàlez, M., The fine spectrum of the Cesàro operator in \(\ell_p (1 < p < \infty )\), Archiv der Mathematik, 44, 4, 355-358 (1985) · Zbl 0568.47021 · doi:10.1007/BF01235779
[2] Cartlidge, P. J., Weighted mean matrices as operators on \(ℓ^p\) [Ph.D. dissertation] (1978), Indiana University
[3] Okutoyi, J. I., On the spectrum of \(C_1\) as an operator on \(b v_0\), Australian Mathematical Society A, 48, 1, 79-86 (1990) · Zbl 0691.40004 · doi:10.1017/S1446788700035205
[4] Okutoyi, J. I., On the spectrum of \(C_1\) as an operator on bv, Communications A, 41, 1-2, 197-207 (1992) · Zbl 0831.47020
[5] Yıldırım, M., On the spectrum of the Rhaly operators on \(\ell_p\), Indian Journal of Pure and Applied Mathematics, 32, 2, 191-198 (2001) · Zbl 0989.47023
[6] 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
[7] Altay, B.; Başar, F., On the fine spectrum of the generalized difference operator \(B(r, s)\) over the sequence spaces \(c_0\) and \(c\), International Journal of Mathematics and Mathematical Sciences, 2005, 18, 3005-3013 (2005) · Zbl 1098.39013 · doi:10.1155/IJMMS.2005.3005
[8] Akhmedov, A. M.; Başar, F., On the fine spectra of the difference operator \(\Delta\) over the sequence space \(\ell_p, (1 \leq p < \infty )\), Demonstratio Mathematica, 39, 3, 585-595 (2006) · Zbl 1118.47303
[9] 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 Mathematica Sinica, 23, 10, 1757-1768 (2007) · Zbl 1134.47025 · doi:10.1007/s10114-005-0777-0
[10] 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
[11] Furkan, H.; Bilgiç, H.; Başar, F., On the fine spectrum of the operator \(B(r, s, t)\) over the sequence spaces \(\ell_p\) and \(b v_p,(1 < p < \infty )\), Computers & Mathematics with Applications, 60, 7, 2141-2152 (2010) · Zbl 1222.47050 · doi:10.1016/j.camwa.2010.07.059
[12] 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
[13] Karaisa, A., Fine spectra of upper triangular double-band matrices over the sequence space \(\ell_p, (1 < p < \infty )\), Discrete Dynamics in Nature and Society, 2012 (2012) · Zbl 1263.47038
[14] Kreyszig, E., Introductory Functional Analysis with Applications (1978), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0368.46014
[15] Appell, J.; Pascale, E.; Vignoli, A., Nonlinear Spectral Theory. Nonlinear Spectral Theory, de Gruyter Series in Nonlinear Analysis and Applications, 10 (2004), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 1056.47001 · doi:10.1515/9783110199260
[16] Goldberg, S., Unbounded Linear Operators (1985), New York, NY, USA: Dover Publications, New York, NY, USA · Zbl 1152.47001
[17] Choudhary, B.; Nanda, S., Functional Analysis with Applications (1989), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0698.46001
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.