Research Article
BibTex RIS Cite
Year 2019, Volume: 68 Issue: 2, 1852 - 1866, 01.08.2019
https://doi.org/10.31801/cfsuasmas.474512

Abstract

References

  • khmedov, A. M. and El-Shabrawy, S. R., On the spectrum of the generalized lower triangle double-band matrices, International Conference on Functional Analysis Proceeding, 17-21 November 2010, Lviv, Ukraine, 2010, 12-13.
  • Akhmedov, A. M. and El-Shabrawy, S. R., Notes on the spectrum of lower triangular double-band matrices, Thai Journal of Mathematics 10, (2012), 415--421.
  • Aurentz, J. L., and Trefethen, L. N., Block operators and spectral discretizations, SIAM Review 59, (2017), 423--446.
  • Baliarsingh, P. and Dutta, S., On a spectral classification of the operator Δ^{r}_{v} over the sequence space c₀, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 84, 2014, 555-561.
  • Böttcher, A. and Grudsky, S., Toeplitz Matrices, Asymptotic Linear Algebra and Functional Analysis, Berlin, Springer-Verlag, 1991.
  • Böttcher, A. and Silbermann, B., Analysis of Teoplitz Operators, Berlin, Springer-Verlag, 1990.
  • Demuth, M., Mathematical aspect of physics with non-selfadjoint operators, List of open problem, American Institute of Mathematics Workshop, 8-12 June 2015.
  • Dunford, N. and Schwartz, J. T., Linear Operators I, New York, Interscience Publishers, 1958.
  • El-Shabrawy, S. R., Spectra and fine spectra of certain lower triangular double-band matrices as operator on c₀, Journal of Inequalities and Applications 1, (2014), 2--9.
  • Gohberg, I. C., Goldberg, S. and Kaashoek, M. A., Basic Classes of Linear Operators, Berlin, Birkhauser, 2003.
  • Gohberg I. C., Krein M. G., Introduction to the Theory of Linear Non-Selfadjoint Operators in Hilbert Space, Rhode Island, USA, American Mathematical Society, 1969.
  • Ismailov, Z. I., Otkun Çevik, E. and Unluyol, E., Compact inverses of multipoint normal differential operators for first order, Electronic Journal of Differential Equations 89, (2011), 1-11.
  • Jeribi A. Spectral Theory and Applications of Linear Operators and Block Operator Matrices, Switzerland, Springer, 2015.
  • Kochubei, A. N., Symmetric operators and nonclassical spectral problems, Matematicheskie Zametki 25, (1979), 425--434.
  • Nagel, R., The spectrum of unbounded operator matrices with non-diagonal domain, Journal of Functional Analysis 89, (1990), 291--302.
  • Naimark, M. A. and Fomin, S. V., Continuous direct sums of Hilbert spaces and some of their applications, Uspekhi Mat Nauk 10, (1955), 111--142 (in Russian).
  • Otkun Çevik, E. and Ismailov, Z. I., Spectrum of the direct sum of operators, Electronic Journal of Differential Equations 210, 2012, 1--8.
  • Pietsch, A., Operators Ideals, North-Holland Publishing Company, Amsterdam, Holland, 1980.
  • Pietsch, A., Eigenvalues and s-Numbers, Cambridge University Press, Londan, England, 1987.
  • Schmidt. E., Zur theorie der linearen und nichtlinearen integralgleichungen Math. Ann. 64, (1907), 433--476.
  • Srivastava, P. D. and Kumar, S., Fine spectrum of the generalized difference operator Δ_{v} on sequence space l₁, Thai Journal of Mathematics 8, (2010), 221--233.
  • Tretter, Ch., Spectral Theory of Block Operator Matrices and Applications, Londan, Imperial CollegePress, 2008.
  • Tripathy, B. C. and Das, R., Spectrum and fine spectrum of the lower triangular matrix B(r,0,s) over the sequences spaces, Appl Math Inf Sci 9, (2015), 2139--2145.
  • von Neumann, J. and Schatten, R., The cross-space of linear transformations, Ann. Math. 47, (1946), 608-630.
  • Zettl, A. ,Sturm-Liouville Theory, USA, American Mathematical Society, 2005.

Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices

Year 2019, Volume: 68 Issue: 2, 1852 - 1866, 01.08.2019
https://doi.org/10.31801/cfsuasmas.474512

Abstract


In this paper, the boundedness and compactness properties of infinite tridiagonal block operator matrices in the direct sum of Hilbert spaces are studied. The necessary and sufficient conditions for these operators belong to Schatten-von Neumann class are given. Then, the results are supported by applications.

References

  • khmedov, A. M. and El-Shabrawy, S. R., On the spectrum of the generalized lower triangle double-band matrices, International Conference on Functional Analysis Proceeding, 17-21 November 2010, Lviv, Ukraine, 2010, 12-13.
  • Akhmedov, A. M. and El-Shabrawy, S. R., Notes on the spectrum of lower triangular double-band matrices, Thai Journal of Mathematics 10, (2012), 415--421.
  • Aurentz, J. L., and Trefethen, L. N., Block operators and spectral discretizations, SIAM Review 59, (2017), 423--446.
  • Baliarsingh, P. and Dutta, S., On a spectral classification of the operator Δ^{r}_{v} over the sequence space c₀, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 84, 2014, 555-561.
  • Böttcher, A. and Grudsky, S., Toeplitz Matrices, Asymptotic Linear Algebra and Functional Analysis, Berlin, Springer-Verlag, 1991.
  • Böttcher, A. and Silbermann, B., Analysis of Teoplitz Operators, Berlin, Springer-Verlag, 1990.
  • Demuth, M., Mathematical aspect of physics with non-selfadjoint operators, List of open problem, American Institute of Mathematics Workshop, 8-12 June 2015.
  • Dunford, N. and Schwartz, J. T., Linear Operators I, New York, Interscience Publishers, 1958.
  • El-Shabrawy, S. R., Spectra and fine spectra of certain lower triangular double-band matrices as operator on c₀, Journal of Inequalities and Applications 1, (2014), 2--9.
  • Gohberg, I. C., Goldberg, S. and Kaashoek, M. A., Basic Classes of Linear Operators, Berlin, Birkhauser, 2003.
  • Gohberg I. C., Krein M. G., Introduction to the Theory of Linear Non-Selfadjoint Operators in Hilbert Space, Rhode Island, USA, American Mathematical Society, 1969.
  • Ismailov, Z. I., Otkun Çevik, E. and Unluyol, E., Compact inverses of multipoint normal differential operators for first order, Electronic Journal of Differential Equations 89, (2011), 1-11.
  • Jeribi A. Spectral Theory and Applications of Linear Operators and Block Operator Matrices, Switzerland, Springer, 2015.
  • Kochubei, A. N., Symmetric operators and nonclassical spectral problems, Matematicheskie Zametki 25, (1979), 425--434.
  • Nagel, R., The spectrum of unbounded operator matrices with non-diagonal domain, Journal of Functional Analysis 89, (1990), 291--302.
  • Naimark, M. A. and Fomin, S. V., Continuous direct sums of Hilbert spaces and some of their applications, Uspekhi Mat Nauk 10, (1955), 111--142 (in Russian).
  • Otkun Çevik, E. and Ismailov, Z. I., Spectrum of the direct sum of operators, Electronic Journal of Differential Equations 210, 2012, 1--8.
  • Pietsch, A., Operators Ideals, North-Holland Publishing Company, Amsterdam, Holland, 1980.
  • Pietsch, A., Eigenvalues and s-Numbers, Cambridge University Press, Londan, England, 1987.
  • Schmidt. E., Zur theorie der linearen und nichtlinearen integralgleichungen Math. Ann. 64, (1907), 433--476.
  • Srivastava, P. D. and Kumar, S., Fine spectrum of the generalized difference operator Δ_{v} on sequence space l₁, Thai Journal of Mathematics 8, (2010), 221--233.
  • Tretter, Ch., Spectral Theory of Block Operator Matrices and Applications, Londan, Imperial CollegePress, 2008.
  • Tripathy, B. C. and Das, R., Spectrum and fine spectrum of the lower triangular matrix B(r,0,s) over the sequences spaces, Appl Math Inf Sci 9, (2015), 2139--2145.
  • von Neumann, J. and Schatten, R., The cross-space of linear transformations, Ann. Math. 47, (1946), 608-630.
  • Zettl, A. ,Sturm-Liouville Theory, USA, American Mathematical Society, 2005.
There are 25 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Review Articles
Authors

Pembe Ipek Al 0000-0002-6111-1121

Zameddin Ismailov 0000-0001-5193-5349

Publication Date August 1, 2019
Submission Date October 24, 2018
Acceptance Date February 12, 2019
Published in Issue Year 2019 Volume: 68 Issue: 2

Cite

APA Ipek Al, P., & Ismailov, Z. (2019). Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1852-1866. https://doi.org/10.31801/cfsuasmas.474512
AMA Ipek Al P, Ismailov Z. Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2019;68(2):1852-1866. doi:10.31801/cfsuasmas.474512
Chicago Ipek Al, Pembe, and Zameddin Ismailov. “Schatten-Von Neumann Characteristic of Infinite Tridiagonal Block Operator Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 2 (August 2019): 1852-66. https://doi.org/10.31801/cfsuasmas.474512.
EndNote Ipek Al P, Ismailov Z (August 1, 2019) Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 2 1852–1866.
IEEE P. Ipek Al and Z. Ismailov, “Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 2, pp. 1852–1866, 2019, doi: 10.31801/cfsuasmas.474512.
ISNAD Ipek Al, Pembe - Ismailov, Zameddin. “Schatten-Von Neumann Characteristic of Infinite Tridiagonal Block Operator Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/2 (August 2019), 1852-1866. https://doi.org/10.31801/cfsuasmas.474512.
JAMA Ipek Al P, Ismailov Z. Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1852–1866.
MLA Ipek Al, Pembe and Zameddin Ismailov. “Schatten-Von Neumann Characteristic of Infinite Tridiagonal Block Operator Matrices”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 2, 2019, pp. 1852-66, doi:10.31801/cfsuasmas.474512.
Vancouver Ipek Al P, Ismailov Z. Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(2):1852-66.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.