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Spectra of Lower Triangular Infinite Matrices Formed by Oscillatory Sequences

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Abstract

We consider an infinite lower triangular matrix, which is a difference operator, with three bands formed by oscillatory sequences. We show that the matrix is a bounded linear operator over the space \(c_0\). We determine the spectrum and its subdivisions of the matrix over \(c_0\). Then, we generalize the matrix to a \((p+1)\)th band infinite lower triangular matrix. We give an estimation of the spectrum of this generalized matrix. Finally, we compare our results with other standard estimations.

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References

  1. Altay, B., Başar, F.: On the fine spectrum of the difference operator \(\bigtriangleup \) on \(c_0\) and \(c\). Inform. Sci. 168(1), 217–224 (2004)

    MathSciNet  MATH  Google Scholar 

  2. 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. 2005(18), 3005–3013 (2005)

    MATH  Google Scholar 

  3. Amirov, R.K., Durna, N., Yildirim, M.: Subdivisions of the spectra for Cesàro, Rhaly and weighted mean operators on \(c_0\), \(c\) and \(l_p\). IJST A3, 175–183 (2011)

    Google Scholar 

  4. Appell, J., De Pascale, E., Vignoli, A.: Nonlinear Spectral Theory, vol. 10. Walter de Gruyter, Berlin (2004)

    Book  MATH  Google Scholar 

  5. Basar, F.: Summability Theory and Its Applications. Bentham Science Publishers, Istanbul (2012)

    Book  MATH  Google Scholar 

  6. Basar, F., Durna, N., Yildirim, M.: Subdivisions of the spectra for generalized difference operator over certain sequence spaces. Thai J. Math. 9(2), 285–295 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Birbonshi, R., Srivastava, P.D.: On some study of the fine spectra of \(n\)-th band triangular matrices. Complex Anal. Oper. Theory 11(4), 739–753 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Dalal, A., Govil, N.K.: On region containing all the zeros of a polynomial. Appl. Math. Comput. 219(17), 9609–9614 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Dündar, E., Başar, F.: On the fine spectrum of the upper triangle double band matrix \(\bigtriangleup ^{+}\) on the sequence space \( c_0\). Math. Commun. 18(2), 337–348 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Durna, N.: Spectra and fine spectra of the upper triangular band matrix U\((a; 0; b)\) on the Hahn sequence space. Math. Commun. 25(1), 49–66 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Durna, N.: Spectra and fine spectra of the upper triangular band matrix U\((a; 0; b)\) over the sequence space \(c_0\). Miskolc Math. Notes 20(1), 209–223 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Durna, N.: Spectra of the upper triangular band matrix U\((r; 0; s)\) on the Hahn space. Bull. Int. Math. Virtual Inst. 10(1), 9–17 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Durna, N., Kilic, R.: Spectra and fine spectra for the upper triangular band matrix U\((a_0, a_1, a_2; b_0, b_1, b_2)\) over the sequence space \(c_0\). Proyecciones (Antofagasta) 38(1), 145–162 (2019)

    MATH  Google Scholar 

  14. El-Shabrawy, S.R.: Spectra and fine spectra of certain lower triangular double-band matrices as operators on \(c_0\). J. Inequal. Appl. 2014(1), 241 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators, vol. 63. Birkhäuser, Basel (2013)

    MATH  Google Scholar 

  16. Gonzalez, M.: The fine spectrum of the Cesàro operator in \(l_p (1<p<\infty )\). Arch. Math. 44(4), 355–358 (1985)

  17. Kreyszig, E.: Introductory Functional Analysis with Applications, vol. 1. Wiley, New York (1989)

    MATH  Google Scholar 

  18. Mursaleen, M., Başar, F.: Sequence Spaces: Topics in Modern Summability Theory. CRC Press, Boca Raton (2020)

    Book  MATH  Google Scholar 

  19. Patra, A., Birbonshi, R., Srivastava, P.D.: On some study of the fine spectra of triangular band matrices. Complex Anal. Oper. Theory 13(3), 615–635 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Patra, A., Srivastava, P.D.: Spectrum and fine spectrum of generalized lower triangular triple band matrices over the sequence space \(l_p\). Linear Multilinear Algebra 68(9), 1797–1816 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Reade, J.B.: On the spectrum of the Cesàro operator. Bull. Lond. Math. Soc. 17(3), 263–267 (1985)

    Article  MATH  Google Scholar 

  22. Rhoades, B., Yildirim, M.: Spectra and fine spectra for factorable matrices. Integral Equ. Oper. Theory 53(1), 127–144 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rhoades, B., Yildirim, M.: Spectra for factorable matrices on \(l_p\). Integral Equ. Oper. Theory 55(1), 111–126 (2006)

    MATH  Google Scholar 

  24. Rhoades, B.E.: The fine spectra for weighted mean operators. Pac. J. Math. 104(1), 219–230 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rhoades, B.E.: The fine spectra for weighted mean operators in B\((l^p)\). Integral Equ. Oper. Theory 12(1), 82–98 (1989)

    MATH  Google Scholar 

  26. Rhoades, B.E., Yildirim, M.: The spectra and fine spectra of factorable matrices on \( c_0\). Math. Commun. 16(1), 265–270 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Srivastava, P.D.: Spectrum and fine spectrum of the generelized difference operators acting on some Banach sequence spaces–a review. Ganita 67(3), 57–68 (2017)

    MathSciNet  Google Scholar 

  28. Stieglitz, M., Tietz, H.: Matrixtransformationen von folgenräumen eine ergebnisübersicht. Math. Z. 154(1), 1–16 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tripathy, B.C., Paul, A.: The spectrum of the operator D\((r,0, s,0, t)\) over the sequence spaces \(c_0\) and \(c\). J. Math. 2013, 1–7 (2013)

    MATH  Google Scholar 

  30. Wenger, R.B.: The fine spectra of the Hölder summability operators. Indian J. Pure Appl. Math. 6(6), 695–712 (1975)

    MathSciNet  MATH  Google Scholar 

  31. Wilansky, A.: Summability Through Functional Analysis. North-Holland Mathematics Studies, Amsterdam (1984)

    MATH  Google Scholar 

  32. Yeşilkayagil, M., Başar, F.: On the fine spectrum of the operator defined by the lambda matrix over the spaces of null and convergent sequences. Abstr. Appl. Anal. 2013, Article ID 687393, 1–13 (2013)

  33. Yeşilkayagil, M., Başar, F.: A survey for the spectrum of triangles over sequence spaces. Numer. Funct. Anal. Optim. 40(16), 1–20 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yildirim, M.: On the spectrum and fine spectrum of the compact Rhaly operators. Indian J. Pure Appl. Math. 27(8), 779–784 (1996)

    MathSciNet  MATH  Google Scholar 

  35. Yildirim, M., Mursaleen, M., Dogan, Ç.: The spectrum and fine spectrum of generalized Rhaly–Cesàro matrices on \(c_0\) and \(c\). Oper. Matrices 12(4), 955–975 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Yildirim, M.E.: The spectrum and fine spectrum of q-Cesàro matrices with \(0< q< 1\) on \(c_0\). Numer. Funct. Anal. Optim. 41(3), 361–377 (2020)

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Acknowledgements

The authors are thankful to the editor and the reviewers for their careful readings and suggestions, which improved the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, 721302, India

    Sanjay Kumar Mahto

  2. Department of Mathematics, R. N. A. R. College Samastipur, LNMU, Darbhanga, Bihar, 848101, India

    Sanjay Kumar Mahto

  3. Department of Mathematics, Indian Institute of Technology Bhilai, Bhilai, Chattisgarh, 492015, India

    P. D. Srivastava

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  1. Sanjay Kumar Mahto
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Correspondence to Sanjay Kumar Mahto.

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Communicated by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.

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This article is part of the topical collection “Linear Operators and Linear Systems” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.

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Mahto, S.K., Srivastava, P.D. Spectra of Lower Triangular Infinite Matrices Formed by Oscillatory Sequences. Complex Anal. Oper. Theory 15, 78 (2021). https://doi.org/10.1007/s11785-021-01131-5

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