Acta mathematica scientia, Series B >
NOTES ON THE SPECTRAL PROPERTIES OF THE WEIGHTED MEAN DIFFERENCE OPERATOR G(u, v;Δ) OVER THE SEQUENCE SPACE l1
Received date: 2014-09-24
Revised date: 2015-01-11
Online published: 2016-04-25
In the study by Baliarsingh and Dutta[Internat. J.Anal., Vol.2014(2014), Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G(u, v;Δ) over the sequence space l1. The product operator G(u, v;Δ) over l1 is defined by (G(u, v;Δ) x)k=
ukvi (xi-xi-1) with xk=0 for all k<0, where x=(xk)∈l1, and u and v are either constant or strictly decreasing sequences of positive real numbers satisfying certain conditions. In this article we give some improvements of the computation of the spectrum of the operator G(u, v;Δ) on the sequence space l1.
Vatan KARAKAYA , Ezgi ERDOGAN . NOTES ON THE SPECTRAL PROPERTIES OF THE WEIGHTED MEAN DIFFERENCE OPERATOR G(u, v;Δ) OVER THE SEQUENCE SPACE l1[J]. Acta mathematica scientia, Series B, 2016 , 36(2) : 477 -486 . DOI: 10.1016/S0252-9602(16)30014-5
[1] Wenger R B. The fine spectra of Holder summability operators. Indian J Pure Appl Math 6, 1975:695-712
[2] Rhoades B E. The fine spectra for weighted mean operators. Pacific J Math, 1983, 104(1):219-230
[3] Gonzales M. The fine spectrum of the Cesaro operator in lp, (1< p< ∞). Arch Math, 1985, 44:355-358
[4] Tripathy B C, Saikia P. On the spectrum of the Cesaro oprator C1 on bv ∩l∞. Math Slovaca, 2013, 63(3):563-572
[5] Co?kun C. The spectra and fine spectra for p-Cesaro operators. Turkish J Math, 1997, 21:207-212
[6] de Malafosse B. Properties of some sets of sequences and application to the spaces of bounded difference sequences of order μ. Hokkaido Math J, 2002, 31:283-299
[7] Altay B, Ba?ar F. On the fine spectrum of the difference operator Δ on c0 and c. Inform Sci, 2004, 168:217-224
[8] Akhmedov A M, Ba?ar F. On the fine spectrum of the Cesaro operator in c0. Math J Ibaraki Univ, 2004, 36:25-32
[9] Akhmedov A M, Ba?ar F. On the spectra of the difference operator Δ over the sequence space lp. Demonstratio Math, 2006, 39(3):585-595
[10] Akhmedov A M, Ba?ar F. On the fine spectra of the difference operator Δ over the sequence space bvp, (1 ≤ p<∞). Acta Math Sin (Engl Ser), 2007, 23(10):1757-1768
[11] Altay B, Ba?ar F. The fine spectrum and the matrix domain of the difference operator Δ on the sequence space lp, (0< p< 1). Commun Math Anal, 2007, 2:1-11
[12] Bilgiç H, Furkan H. On the fine spectrum of generalized difference operator B(r, s) over the sequence spaces lp and bvp. Nonlinear Anal, 2008, 68:499-506
[13] Tripathy B C, Paul A. Spectra of the operator B (f, g) on the vector valued sequence space c0(X). Bol Soc Parana Mat (3), 2013, 31(1):105-111
[14] Karakaya V, Manafov M Dzh, ?im?ek N. On the fine spectrum of the second order difference operator over the sequence spaces lp and bvp. Math Comput Modelling, 2012, 55(3):426-436
[15] Altun M, Karakaya V. Fine spectra of lacunary matrices. J Commun Math Anal, 2009, 7(1):1-10
[16] Karakaya V, Altun M. Fine spectra of upper triangular double-band matrices. J Comput Appl Math, 2010, 234:1387-1394
[17] Durna N, Y?ld?r?m M. Subdivision of the spectra for factorable matrices on c and lp. Math Commun, 2011, 16:519-530
[18] Rhoades B E, Y?ld?r?m M. The spectra and fine spectra of factorable matrices on c0. Math Commun, 2011, 16:265-270
[19] Ba?ar F. Summability Theory and Its Applications. Bentham Science Publishers, e-books, Monographs, Istanbul, 2012
[20] Tripathy B C, Das R. Spectra of the Rhaly operator on the sequence space bv0∩l∞. Bol Soc Parana Mat (3), 2014, 32(1):263-275
[21] Dündar E, Ba?ar F. On the fine spectrum of the upper triangle double band matrix Δ+ on the sequence space c0. Math Commun, 2013, 18:337-348
[22] Karaisa A, Ba?ar F. Fine spectra of the upper triangular triple band matrices over the sequence space lp, (0< p< ∞). Abstr Appl Anal, 2013, 2013. ID 342682
[23] Erdo?an E, Karakaya V. On spectral properties of a new operator over sequence spaces c and c0. Acta Math Sci Ser B Engl Ed, 2014, 34(5):1481-1494
[24] Kreyszig E. Introductory Functional Analysis with Applications. New York:John Wiley and Sons Inc, 1978
[25] Appell J, Pascale E, Vignoli A. Nonlinear Spectral Theory. de Gruyter Ser Nonlinear Anal Appl, Walter de Gruyter·Berlin·New York, 2004
[26] Wilansky A. Summability Through Functional Analysis. North-Holland Mathematics Studies, Vol 85. Amsterdam:North Holland, 1984
[27] Goldberg S. Unbounded Linear Operators. Mc Graw-Hill Book Comp, 1966
[28] Baliarsingh P, Dutta S. On the spectral properties of the weighted mean difference operator G(u, v;Δ) over the sequence space l1. Internat J Anal, 2014, 2014
[29] Polat H, Karakaya V and ?im?ek N. Difference sequences spaces derived by using a generalized weighted mean. Appl Math Lett, 2011, 24:608-614
/
| 〈 |
|
〉 |