# Some Remarks on the Matrix Domain and the Spectra of a Generalized Difference Operator

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## Abstract

Recently, Meng and Mei (J Math Anal Appl, 2018. https://doi.org/10.1016/j.jmaa.2018.10.051) have introduced two Euler difference sequence spaces \(e_0^t(B_\nu ^{(m)})\) and \(e_c^t(B_\nu ^{(m)})\) using generalized difference operator \(B_\nu ^{(m)}\). Some functional properties and Köthe–Toeplitz duals of the proposed spaces have been determined. Also, the fine spectra of the generalized difference operator \(B_\nu ^{(m)}\) over the sequence space \(\ell _1\) have been computed following the results of Dutta and Baliarsingh (Appl Math Comput 219:1776–1784, 2012). In fact, the results for the spectra of the said difference operators are new and interesting. Upon further investigations, it has been noticed that these results should be improved and modified. The primary objective of this note is to provide such sorts of substantial modifications on these results.

## Keywords

The difference operators \(\Delta _\nu ^{m}\) and \(B_\nu ^{(m)}\) Difference sequence spaces The spectrum of an operator## Mathematics Subject Classification

47A10 40A05 46A45## Notes

### Compliance with Ethical Standards

### Conflict of interest

The author declares that they have no conflict of interest.

## References

- Akhmedov AM, Başar F (2006) The fine spectra of the difference operator \(\Delta\) over the sequence space \(\ell _p, (1\le p < \infty )\). Demonstr Math 39:586–595Google Scholar
- Akhmedov AM, Başar F (2007) On the fine spectra of the difference operator \(\Delta\) over the sequence space \(bv_p, (1\le p < \infty )\). Acta Math Sin (Engl) Ser 23(10):1757–1768MathSciNetCrossRefGoogle Scholar
- Akhmedov AM, Başar F (2008) The fine spectra of the Cesaro operator \(C_1\) over the sequence space \(bv_p\), \((1\le p < \infty )\). Math J Okayama Univ 50:135–147MathSciNetzbMATHGoogle Scholar
- Akhmedov AM, El-shabrawy SR (2011) On the fine spectrum of the operator \(\Delta _{a, b}\) over the sequence space \(c\). Comput Math Appl 61:29943002MathSciNetCrossRefGoogle Scholar
- Altay B, Başar F (2004) On the fine spectrum of the difference operator on \(c_0\) and \(c\). Inf Sci 168:217–224CrossRefGoogle Scholar
- Altay B, Başar F (2005) On the fine spectrum of the generalized difference operator \(B(r, s)\) over the sequence space \(c_0\) and \(c\). Int Math Math Sci 18:3005–3013CrossRefGoogle Scholar
- Altay B, Başar F (2007) The fine spectrum and the matrix domain of the difference operator \(\Delta\) on the sequence space \(\ell _p, (0 < p < 1)\). Commun Math Anal 2:1–11MathSciNetzbMATHGoogle Scholar
- Altay B, Karakuş M (2005) On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces. Thai J Math 3:153–162MathSciNetzbMATHGoogle Scholar
- Baliarsingh P (2013) Some new difference sequence spaces of fractional order and their dual spaces. Appl Math Comput 219(18):9737–9742MathSciNetzbMATHGoogle Scholar
- Baliarsingh P (2016) On a fractional difference operator. Alex Eng J 55(2):1811–1816CrossRefGoogle Scholar
- Baliarsingh P, Dutta S (2014) On a spectral classification of the operator \(\Delta _\nu ^ r\) over the Sequence Space \(c_0\). Proc Natl Acad Sci India Sect A Phys Sci 84(4):555–561CrossRefGoogle Scholar
- Baliarsingh P, Dutta S (2015a) A unifying approach to the difference operators and their applications. Bol Soc Paran Mat 33(1):49–57MathSciNetCrossRefGoogle Scholar
- Baliarsingh P, Dutta S (2015b) On the classes of fractional order difference sequence spaces and their matrix transformations. Appl Math Comput 250:665–674MathSciNetzbMATHGoogle Scholar
- Baliarsingh P, Dutta S (2016) On certain Toeplitz matrices via difference operator and their applications. Afrika Mat 27(5–6):781–793MathSciNetCrossRefGoogle Scholar
- Başar F (2012) Summability theory and its applications. Bentham Science Publishers. Istanbul. eISBN: 978-160805-252Google Scholar
- Başarır M , Kayıkçı M (2009) On the generalized \(B^m\)-Riesz difference sequence space and \(\beta\)-property. J. Inequal Appl 2009(1)Google Scholar
- Başarır M, Kara E (2011) On compact operators on the Riesz \(B^m\)-difference sequence spaces. Iran J Sci Technol Trans A Sci 35:279–285MathSciNetzbMATHGoogle Scholar
- Bilgiç H, Furkan H, Altay B (2007) On the fine spectrum of the operator \(B(r, s, t)\) over \(c_0\) and \(c\). Comput Math Appl 53:989–998MathSciNetCrossRefGoogle Scholar
- Dündar E, Başar F (2013) On the fine spectrum of the upper triangle double band matrix \(\Delta ^+\) on the sequence space \(c_0\). Math Commun 18:337–348MathSciNetzbMATHGoogle Scholar
- Dutta S, Baliarsingh P (2012) On the fine spectra of the generalized rth difference operator \(\Delta _\nu ^r\) on the sequence space \(\ell _1\). Appl Math Comput 219:1776–1784MathSciNetzbMATHGoogle Scholar
- Dutta S, Baliarsingh P (2013a) On the spectrum of 2-nd order generalized difference operator \(\Delta ^2\) over the sequence space \(c_0\). Bol Soc Paran Mat 31(2):235–244CrossRefGoogle Scholar
- Dutta S, Baliarsingh P (2013b) Some spectral aspects of the operator \(\Delta _\nu ^r\) v over the sequence spaces \(\ell _p\) and \(bv_p, (1 <p < \infty )\). Chin J Math 2013:1–10Google Scholar
- Et M, Basarır M (1997) On some new generalized difference sequence spaces. Period Math Hungar 35(3):169–175MathSciNetCrossRefGoogle Scholar
- Furkan H, Bilgiç H, Kayaduman K (2006) On the fine spectrum of the generalized difference operator \(B(r, s)\) over the sequence spaces \(\ell _1\) and \(bv\). Hokkaido Math J 35(4):893–904MathSciNetCrossRefGoogle Scholar
- Karaisa A, Başar F (2013) Fine spectra of upper triangular triple-band matrices over the sequence space \(\ell _p,\; (0 < p <\infty )\). Abstr Appl Anal 2013, 342682zbMATHGoogle Scholar
- Karaisa A, Başar F (2015) On the fine spectrum of the generalized difference operator defined by a double sequential band matrix over the sequence space \(\ell _p, (1 < p < \infty )\). Hacet J Math Stat 44(6):1315–1332MathSciNetzbMATHGoogle Scholar
- Kayaduman K, Furkan H (2006) The fine spectra of the difference operator \(\Delta\) over the sequence spaces \(\ell _1\) and \(bv\). Int Math Forum 1(24):1153–1160MathSciNetCrossRefGoogle Scholar
- Meng J, Mei L (2018) The matrix domain and the spectra of a generalized difference operator. J Math Anal Appl. https://doi.org/10.1016/j.jmaa.2018.10.051 CrossRefGoogle Scholar
- Mursaleen M, Yildirim M, Durna N (2019) On the spectrum and Hilbert Schimidt properties of generalized Rhaly matrices. Commun Fac Sci Univ Ank Ser A1 68(1):712–723MathSciNetGoogle Scholar
- Panigrahi B, Srivastava PD (2011) Spectrum and fine spectrum of the generalized second order difference operator \(\Delta _{uv}^2\) on sequence space \(c_0\). Thai J Math 9:57–74MathSciNetGoogle Scholar
- Panigrahi B, Srivastava PD (2012) Spectrum and fine spectrum of the generalized second order forward difference operator \(\Delta _{uvw}^2\) on sequence space \(\ell _1\). Demonstr Math 45:593–609Google Scholar
- Srivastava PD, Kumar S (2009) On the fine spectrum of the generalized difference operator \(\Delta _\nu\) over the sequence space \(c_0\). Commun Math Anal 6(1):8–21MathSciNetGoogle Scholar
- Yeşilkayagil M, Başar F (2013) On the fine spectrum of the operator defined by a lambda matrix over the sequence spaces of null and convergent sequences. Abstr Appl Anal 2013, 687393MathSciNetCrossRefGoogle Scholar
- Yildirim M, Mursaleen M, Doğn Ç (2018) The spectrum and fine spectrum of generalized Rhaly-Cesàro matrices on \(c_0\) and \(c\). Oper Matrices 12(4):955–975MathSciNetCrossRefGoogle Scholar