Skip to main content
SpringerLink
Log in
Book cover

Applied Mathematical Analysis: Theory, Methods, and Applications pp 791–810Cite as

On the Spectra of Difference Operators Over Some Banach Spaces

  • Chapter
  • First Online:

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 177))

Abstract

The primary objective of this chapter is to provide a literature review on spectral subdivisions of difference operators and compute the spectrum and the fine spectrum of third order difference operator \(\Delta ^3\) over the Banach space c. The generalized difference operator \(\Delta ^3:c\rightarrow c\) is defined by

$$(\Delta ^3 x)_k=\sum _{i=0}^3(-1)^i\left( {\begin{array}{c}3\\ i\end{array}}\right) x_{k-i}=x_k-3x_{k-1}+3x_{k-2}-x_{k-3}, ~(k\in \mathbb {N}_0)$$

It is presumed that \(x_n=0\) if \(n<0.\) The operator \(\Delta ^3\) represents a lower forth band infinite matrix. Finally, we find the estimates for the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the above operator over the Banach spaces \(c,c_0\) and \(\ell _1\).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Goldberg, S.: Unbounded linear operators. Dover Publications, Inc., New York (1985)

    MATH  Google Scholar 

  2. Kreyszig, E.: Introductory Functional Analysis with Applications. John Wiley and Sons Inc., New York-Chichester -Brisbane-Toronto (1978)

    MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Okutoyi, J.T.: On the spectrum of \(C_1\) as an operator on \(bv\). Commun. Fac. Sci. Univ. Ank. Ser. \(A_1 41\), 197–207 (1992)

    Google Scholar 

  8. Akhmedov, A.M., Başar, F.: On the spectrum of the Cesàro operator in \(c_0\). Math. J. Ibaraki Univ. 36, 25–32 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Akhmedov, A.M., Başar, F.: The fine spectra of Cesàro operator \(C_1\) over the sequence space \(bv_p\). Math. J. Okayama Univ. 50, 135–147 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Akhmedov, A.M., Başar, F.: The fine spectra of the difference operator \(\Delta \) over the sequence space \(\ell _p, (1\le p\,<\,\infty )\). Demonstratio Math. 39(3), 586–595 (2006)

    Google Scholar 

  11. Akhmedov, A.M., Başar, F.: On the fine spectra of the difference operator \(\Delta \) over the sequence space \(bv_p, (1\le p\,<\,\infty )\). Acta. Math. Sin. Engl. ser. Oct. 23(10), 1757–1768 (2007)

    Google Scholar 

  12. Altay, B., Başar, F.: The fine spectrum and the matrix domain of the difference operator \(\Delta \) on the sequence space \(\ell _p, (0\,<\,p\,<\,1)\). Comm. Math. Anal. 2, 1–11 (2007)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Furkan, H., Bilgiç, H., Kayaduman, K.: 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), 893–904 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Kayaduman, K., Furkan, H.: The fine spectra of the difference operator \(\Delta \) over the sequence spaces \(\ell _1\) and \(bv\). Int. Math. For. 1(24), 1153–1160 (2006)

    MATH  Google Scholar 

  16. Srivastava, P.D., Kumar, S.: On the fine spectrum of the generalized difference operator \(\Delta _\nu \) over the sequence space \(c_0\). Commun. Math. Anal. 6(1), 8–21 (2009)

    MathSciNet  Google Scholar 

  17. Akhmedov, A.M., El-Shabrawy, S.R.: On the fine spectrum of the operator \(\Delta _{a, b}\) over the sequence space \(c\). Comput. Math. Appl. 61, 2994–3002 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Baliarsingh, S. Dutta, On a spectral classification of the operator \(\Delta _\nu ^ r\) over the Sequence Space \(c_0\), Proc. Natl. Acad. Sci. India Series A 84(4) (2014) 555–561

    Google Scholar 

  19. Dutta, S., Baliarsingh, P.: On the fine spectra of the generalized rth difference operator \(\Delta _\nu ^r\) on the sequence space \(\ell _1\). Appl. Math. Comput. 219, 1776–1784 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Kızmaz, H.: On Certain Sequence spaces. Canad. Math. Bull. 24(2), 169–176 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Et, M., Çolak, R.: On some generalized difference sequence spaces. Soochow J. Math. 21(4), 377–386 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Baliarsingh, P.: Some new difference sequence spaces of fractional order and their dual spaces. Appl. Math. Comput. 219(18), 9737–9742 (2013)

    MathSciNet  MATH  Google Scholar 

  23. P. Baliarsingh, S. Dutta, A unifying approach to the difference operators and their applications, Bol. Soc. Paran. Mat. 33(1) (2015) 49-57

    Article  MathSciNet  Google Scholar 

  24. Baliarsingh, P., Dutta, S.: On the classes of fractional order difference sequence spaces and their matrix transformations. Appl. Math. Comput. 250, 665–674 (2015)

    MathSciNet  MATH  Google Scholar 

  25. P. Baliarsingh, L. Nayak, A note on fractional difference operators, Alexandria Eng. J. (2017) dio.org/10.1016/j.aej.2017.02.022

    Google Scholar 

  26. P. Baliarsingh, On a fractional difference operator, Alexandria Eng. J. 55(2) (2016) 1811-1816

    Article  Google Scholar 

  27. P. Baliarsingh, H. Dutta, On difference operators and their applications, M. Ruzhansky and H. Dutta (Eds), Advanced topic in Matematical analysis, CRC press, 2018, 405–425

    Google Scholar 

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dutta, A.J., Tripathy, B.C.: Fine spectrum of the generalized difference operator \(B(r s)\) over the class of convergent series. Inter. J. Anal. 4, Art. ID 630436

    Google Scholar 

  30. Et, M., Basarir, M.: On some new generalized difference sequence spaces. Periodica Math. Hungar. 35(3), 169–175 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Srivastava, P.D., Kumar, S.: Fine spectrum of the generalized difference operator \(\Delta _\nu \) on sequence space \(\ell _1\). Appl. Math. Comput. 218(11), 6407–6414 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Wilansky, A.: Summability through Functional Analysis. North-Holland Mathematics Studies, Amsterdam, New York, Oxford (1984)

    MATH  Google Scholar 

  33. Dutta, S., Baliarsingh, P.: On the spectrum of 2-nd order generalized difference operator \(\Delta ^2\) over the sequence space \(c_0\). Bol. Soc. Paran. Mat. 31(2), 235–244 (2013)

    Google Scholar 

  34. Baliarsingh, P.: On a spectral subdivision of the operator \(\Delta _i^2\) over the sequence spaces \(c_0\) and \(\ell _1\). Thai. J, Math (2018). in press

    Google Scholar 

  35. 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

    Google Scholar 

  36. H. Bilgiç, H. Furkan, B. Altay, On the fine spectrum of the operator \(B(r, s, t)\) over \(c_0\) and \(c\), Comput. Math. Appl., 53 (2007) 989-998

    Google Scholar 

  37. Bilgiç, H., Furkan, H., Altay, B.: On the fine spectrum of the operator \(B(r, s, t)\) over the sequence spaces \(c_0\) and \(c\). Math Comput. model. 45, 883–891 (2007)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, 781014, India

    Hemen Dutta

  2. Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar, 751024, India

    P. Baliarsingh

Authors
  1. Hemen Dutta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Baliarsingh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Baliarsingh .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, India

    Hemen Dutta

  2. Computational Intelligence Laboratory, Department of Electrical & Computer Engineering, University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dutta, H., Baliarsingh, P. (2020). On the Spectra of Difference Operators Over Some Banach Spaces. In: Dutta, H., Peters, J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-99918-0_23

Download citation

Publish with us

Policies and ethics

Search

Navigation