Schatten-von Neumann characteristic of infinite tridiagonal block operator matrices

Yıl 2019, Cilt: 68 Sayı: 2, 1852 - 1866, 01.08.2019

Pembe Ipek Al , Zameddin Ismailov

https://doi.org/10.31801/cfsuasmas.474512

Öz

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.

Anahtar Kelimeler

Direct sum of Hilbert spaces, infinite tridiagonal block operator matrices, compact operator, Schatten-von Neumann classes

Kaynakça

• 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.