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Year 2023, Volume: 44 Issue: 1, 181 - 187, 26.03.2023

Nuh Durna , Ömer Özdemir

https://doi.org/10.17776/csj.1209529

Abstract

Keywords

Spectra fine spectra generalized difference operator band matrix

References

  • [1] Kızmaz H., On certain sequence spaces, Can. Math. Bull., 24(2) (1981) 169–176.
  • [2] Et M., Çolak R., On some generalized difference sequence spaces, Soochow J. Math., 21(4) (1995) 377–386.
  • [3] Baliarsingh P., Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput., 219(18) (2013) 9737–9742.
  • [4] Bektaş C.A., Et M., Çolak R., Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl., 292 (2004) 423–432.
  • [5] Et M., Başarır M., On some new generalized difference sequence spaces, Periodica Math. Hung., 35(3) (1997) 169–175.
  • [6] Malkowsky E., Mursaleen M., Suantai S., The dual spaces of sets of difference sequence sequence spaces of order m and matrix transformations, Acta. Math. Sin. (Engl. Ser.), 23(3) (2007) 521–532.
  • [7] Baliarsingh P., Mursalen M., Rakocevic V., A survey on the spectra of the difference operators over the Banach space, c. RACSAM, 115:57. 84(4) (2021) 1–17.
  • [8] Altay B., Başar F., On the fine spectrum of the difference operator on c_0 and c, Inf. Sci., 168 (2004) 217–224.
  • [9] Altay B., Başar F., The fine spectrum and the matrix domain of the difference operator Δ on the sequence space l_p, (1<p<∞), Commun. Math. Anal., 2(2) (2007) 1–11.
  • [10]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 (2005) 3005–3013.
  • [11] Akhmedov A.M., Başar F., The fine spectra of the difference operator Δ over the sequence space bv_p, (1≤p<∞), Demonstr. Math., 39(3) (2006) 586–595.
  • [12] Akhmedov A.M., Başar F., On the fine spectra of the difference operator Δ over the sequence space l_p, (1≤p<∞), Acta. Math. Sin. (Engl. Ser.), 23(10) (2007) 1757–1768.
  • [13] Srivastava P.D., Kumar S., On the fine spectrum of the generalized difference operator Δ_v over the sequence space c_0, Commun. Math. Anal., 6(1) (2009) 8–21.
  • [14] Srivastava P.D., Kumar S., Fine spectrum of the generalized difference operator Δ_v on sequence space l_1, Thai. J. Math., 8(2) (2010) 221–233.
  • [15] Akhmedov A.M., El-Shabrawy S.R., On the fine spectrum of the operator Δ_(a,b) over the sequence space c, Comput. Math. Appl., 61 (2011) 2994–3002.
  • [16] Dutta S., Baliarsingh P., On the fine spectra of the generalized rth difference operator Δ_v^r on the sequence space l_1, Appl. Math. Comput., 219 (2012) 1776–1784.
  • [17] Dutta S., Baliarsingh P., On the spectrum of 2-nd order generalized difference operator Δ^2 over the sequence space c_0, Bol. Soc. Paran. Mat., 31(2) (2013) 235–244.
  • [18] Durna N., Yildirim M., Subdivision of the spectra for factorable matrices on c_0, GUJ Sci., 24 (1) (2011) 45-49.
  • [19] Başar F., Durna N., Yildirim M., Subdivisions of the spectra for genarilized difference operator over certain sequence spaces, Thai J. Math., 9(2) (2011) 285–295.
  • [20] Durna N., Subdivision of the spectra for the generalized upper triangular double-band matrices Δ^uv over the sequence spaces c_0 and c, ADYU Sci., 6(1) (2016), 31-43.
  • [21] Das R., On the spectrum and fine spectrum of the upper triangular matrix U(r_1,r_2;s_1,s_2) over the sequence space c_0, Afr. Math., 28 (2017) 841-849.
  • [22] El-Shabrawy S.R., Abu-Janah S.H., Spectra of the generalized difference operator on the sequence spaces bv_0 and h, Linear Multilinear Algebra, 66(8) (2018) 1691–1708.
  • [23] Tripathy B.C., Das R., Fine spectrum of the upper triangular matrix U(r,0,0,s) over the squence spaces c_0 and c, Proyecciones J. Math., 37(1) (2018), 85-101.
  • [24] Durna N., Yildirim M., Kılıç R., Partition of the spectra for the generalized difference operator B(r,s) on thesequence space cs, Cumhuriyet Sci. J., 39(1) (2018) 7–15.
  • [25] Furkan H., Bilgiç H., Altay B., On the fine spectrum of the operator B(r,s,t) over c_0 and c, Comput. Math. Appl., 53(6) (2007) 989–998.
  • [26] Wilansky A., Summability Through Functional Analysis. North-Holland Mathematics Studies, Amsterdam (1984).
  • [27] Goldberg S., Unbounded Linear Operators, New York: McGraw Hill, (1966).
  • [28] Appell J., Pascale E.D., Vignoli A., Nonlinear Spectral Theory, New York Berlin: Walter de Gruyter, (2004).

Spectral Decompositions of the Difference Operator Δ^m over the Sequence Space cs

Year 2023, Volume: 44 Issue: 1, 181 - 187, 26.03.2023

Nuh Durna , Ömer Özdemir

https://doi.org/10.17776/csj.1209529

Abstract

This study aims to bring together some studies on the spectra of difference operators in the literature over the cs sequence space and to provide a basis for related problems. So far, the problem has been solved up to a maximum of 2 orders on the sequence space cs. In this article, we discuss the difference operator Δ^m, represented by a (m+1) banded matrix, which generalizes the difference operators of the form Δ, Δ^2, B(r,s) and B(r,s,t) and we will give its boundedness, spectrum, fine spectrum and some spectral separations over the sequence space cs.

Keywords

Spectra fine spectra generalized difference operator band matrix

References

  • [1] Kızmaz H., On certain sequence spaces, Can. Math. Bull., 24(2) (1981) 169–176.
  • [2] Et M., Çolak R., On some generalized difference sequence spaces, Soochow J. Math., 21(4) (1995) 377–386.
  • [3] Baliarsingh P., Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput., 219(18) (2013) 9737–9742.
  • [4] Bektaş C.A., Et M., Çolak R., Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl., 292 (2004) 423–432.
  • [5] Et M., Başarır M., On some new generalized difference sequence spaces, Periodica Math. Hung., 35(3) (1997) 169–175.
  • [6] Malkowsky E., Mursaleen M., Suantai S., The dual spaces of sets of difference sequence sequence spaces of order m and matrix transformations, Acta. Math. Sin. (Engl. Ser.), 23(3) (2007) 521–532.
  • [7] Baliarsingh P., Mursalen M., Rakocevic V., A survey on the spectra of the difference operators over the Banach space, c. RACSAM, 115:57. 84(4) (2021) 1–17.
  • [8] Altay B., Başar F., On the fine spectrum of the difference operator on c_0 and c, Inf. Sci., 168 (2004) 217–224.
  • [9] Altay B., Başar F., The fine spectrum and the matrix domain of the difference operator Δ on the sequence space l_p, (1<p<∞), Commun. Math. Anal., 2(2) (2007) 1–11.
  • [10]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 (2005) 3005–3013.
  • [11] Akhmedov A.M., Başar F., The fine spectra of the difference operator Δ over the sequence space bv_p, (1≤p<∞), Demonstr. Math., 39(3) (2006) 586–595.
  • [12] Akhmedov A.M., Başar F., On the fine spectra of the difference operator Δ over the sequence space l_p, (1≤p<∞), Acta. Math. Sin. (Engl. Ser.), 23(10) (2007) 1757–1768.
  • [13] Srivastava P.D., Kumar S., On the fine spectrum of the generalized difference operator Δ_v over the sequence space c_0, Commun. Math. Anal., 6(1) (2009) 8–21.
  • [14] Srivastava P.D., Kumar S., Fine spectrum of the generalized difference operator Δ_v on sequence space l_1, Thai. J. Math., 8(2) (2010) 221–233.
  • [15] Akhmedov A.M., El-Shabrawy S.R., On the fine spectrum of the operator Δ_(a,b) over the sequence space c, Comput. Math. Appl., 61 (2011) 2994–3002.
  • [16] Dutta S., Baliarsingh P., On the fine spectra of the generalized rth difference operator Δ_v^r on the sequence space l_1, Appl. Math. Comput., 219 (2012) 1776–1784.
  • [17] Dutta S., Baliarsingh P., On the spectrum of 2-nd order generalized difference operator Δ^2 over the sequence space c_0, Bol. Soc. Paran. Mat., 31(2) (2013) 235–244.
  • [18] Durna N., Yildirim M., Subdivision of the spectra for factorable matrices on c_0, GUJ Sci., 24 (1) (2011) 45-49.
  • [19] Başar F., Durna N., Yildirim M., Subdivisions of the spectra for genarilized difference operator over certain sequence spaces, Thai J. Math., 9(2) (2011) 285–295.
  • [20] Durna N., Subdivision of the spectra for the generalized upper triangular double-band matrices Δ^uv over the sequence spaces c_0 and c, ADYU Sci., 6(1) (2016), 31-43.
  • [21] Das R., On the spectrum and fine spectrum of the upper triangular matrix U(r_1,r_2;s_1,s_2) over the sequence space c_0, Afr. Math., 28 (2017) 841-849.
  • [22] El-Shabrawy S.R., Abu-Janah S.H., Spectra of the generalized difference operator on the sequence spaces bv_0 and h, Linear Multilinear Algebra, 66(8) (2018) 1691–1708.
  • [23] Tripathy B.C., Das R., Fine spectrum of the upper triangular matrix U(r,0,0,s) over the squence spaces c_0 and c, Proyecciones J. Math., 37(1) (2018), 85-101.
  • [24] Durna N., Yildirim M., Kılıç R., Partition of the spectra for the generalized difference operator B(r,s) on thesequence space cs, Cumhuriyet Sci. J., 39(1) (2018) 7–15.
  • [25] Furkan H., Bilgiç H., Altay B., On the fine spectrum of the operator B(r,s,t) over c_0 and c, Comput. Math. Appl., 53(6) (2007) 989–998.
  • [26] Wilansky A., Summability Through Functional Analysis. North-Holland Mathematics Studies, Amsterdam (1984).
  • [27] Goldberg S., Unbounded Linear Operators, New York: McGraw Hill, (1966).
  • [28] Appell J., Pascale E.D., Vignoli A., Nonlinear Spectral Theory, New York Berlin: Walter de Gruyter, (2004).
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Nuh Durna 0000-0001-5469-7745

Ömer Özdemir 0000-0003-1158-5553

Publication Date March 26, 2023
Submission Date November 24, 2022
Acceptance Date March 16, 2023
Published in Issue Year 2023Volume: 44 Issue: 1

Cite

APA Durna, N., & Özdemir, Ö. (2023). Spectral Decompositions of the Difference Operator Δ^m over the Sequence Space cs. Cumhuriyet Science Journal, 44(1), 181-187. https://doi.org/10.17776/csj.1209529
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