Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University2072610.14498/vsgtu1275Research ArticleInvestigations of the numerical range of a operator matrixRasulovTulkin Kh(Cand. Phys. & Math. Sci.), Assotiate Professor, Dept. of Mathematical Physics & Analysisrth@mail.ruDilmurodovElyor BAssistant Lecturer, Dept. of Mathematical Physics & Analysiselyor.dilmurodov@mail.ruBukhara State University15062014182506318022020Copyright © 2014, Samara State Technical University2014We consider a $2\times2$ operator matrix $A$ (so-called generalized Friedrichs model) associated with a system of at most two quantum particles on ${\rm d}$-dimensional lattice. This operator matrix acts in the direct sum of zero- and one-particle subspaces of a Fock space. We investigate the structure of the closure of the numerical range $W(A)$ of this operator in detail by terms of its matrix entries for all dimensions of the torus ${\bf T}^{\rm d}$. Moreover, we study the cases when the set $W(A)$ is closed and give necessary and sufficient conditions under which the spectrum of $A$ coincides with its numerical range.operator matrixgeneralized Friedrichs modelFock spacenumerical rangepoint and approximate point spectraannihilation and creation operatorsfirst Schur complimentоператорная матрицаобобщённая модель Фридрихсапространство Фокачисловая область значенийточечный и аппроксимативно точечный спектрыоператоры рождения и уничтоженияпервый комплимент Шура[O. Toeplitz, “Das algebraische Analogon zu einem Satze von Fejér” // Math. Z., 1918. vol. 2, no. 1-2. pp. 187-197. doi: 10.1007/BF01212904.][F. Hausdorff, “Der Wertvorrat einer Bilinearform” // Math. Z., 1919. vol. 3, no. 1. pp. 314-316. doi: 10.1007/BF01292610.][A. Wintner, “Zur Theorie der beschränkten Bilinearformen” // Math. Z., 1929. vol. 30, no. 1. pp. 228-281. doi: 10.1007/BF01187766.][H. Langer, A. S. Markus, V. I. Matsaev, C. Tretter, “A new concept for block operator matrices: the quadratic numerical range” // Linear Algebra Appl., 2001. vol. 330, no. 1-3. pp. 89-112. doi: 10.1016/S0024-3795(01)00230-0.][C. Tretter, M. Wagenhofer, “The block numerical range of an n × n block operator matrix” // SIAM. J. Matrix Anal. & Appl., 2003. vol. 24, no. 4. pp. 1003-1017. doi: 10.1137/S0895479801394076.][L. Rodman, I. M. Spitkovsky, “Ratio numerical ranges of operators” // Integr. Equ. Oper. Theory, 2011. vol. 71, no. 2. pp. 245-257. doi: 10.1007/s00020-011-1898-8.][W. S. Cheung, X. Liu, T. Y. Tam, “Multiplicities, boundary points and joint numerical ranges” // Operators and Matrices, 2011. vol. 5, no. 1. pp. 41-52. doi: 10.7153/oam-05-02.][H. L. Gau , C. K. Li, Y. T. Poon, N. S. Sze, “Higher rank numerical ranges of normal matrices” // SIAM. J. Matrix Anal. & Appl., 2011. vol. 32. pp. 23-43, arXiv: 0902.4869 [math.FA]. doi: 10.1137/09076430X.][B. Kuzma, C. K. Li, L. Rodman, “Tracial numerical range and linear dependence of operators” // Electronic J. Linear Algebra, 2011. vol. 22. pp. 22-52. http://eudml.org/doc/223236.][C. K. Li, Y. T. Poon, “Generalized numerical range and quantum error correction” // J. Operator Theory, 2011. vol. 66, no. 2. pp. 335-351. http://www.mathjournals.org/jot/2011-066-002/2011-066-002-004.html.][K. Gustafson, D. K. M. Rao, Numerical range: The field of values of linear operators and matrices, Berlin, Springer, 1997, xiv+190 pp.][D. S. Keeler, L. Rodman, I. M. Spitkovsky, “The numerical range of 3 × 3 matrices” // Linear Algebra and its Appl., 1997. vol. 252, no. 1-3. pp. 115-139. doi: 10.1016/0024-3795(95)00674-5.][H.-L. Gau, “Elliptic numerical range of 4 × 4 matrices” // Taiwanese J. Math., 2006. vol. 10, no. 1. pp. 117-128.][D. Henrion, “Semidefinite geometry of the numerical range” // Electronic J. Linear Algebra, 2010. vol. 20. pp. 322-332. http://eudml.org/doc/229710, arXiv: 0812.1624 [math.OC].][Р. А. Минлос, Я. Г. Синай, “Исследование спектров стохастических операторов, возникающих в решетчатых моделях газа” // ТМФ, 1970. Т. 2, № 2. С. 230-243.][R. A. Minlos, Ya. G. Sinai, “Spectra of stochastic operators arising in lattice models of a gas” // Theoret. and Math. Phys., 1970. vol. 2, no. 2. pp. 167-176. doi: 10.1007/BF01036789.][М. А. Лаврентьев, Б. В. Шабат, Проблемы гидродинамики и их математические модели. М.: Наука, 1973. 416 с.][A. I. Mogilner, “Hamiltonians in solid state physics as multiparticle discrete Schrödinger operators: problems and results” // Adv. Soviet Math., 1991. vol. 5. pp. 139-194.][M. Reed, B. Simon, Methods of modern mathematical physics, V. IV, Analysis of operators, New York-London, Academic Press, 1978, 396 pp.][М. Рид, Б. Саймон, Методы современной математической физики. Т. 4: Анализ операторов. М.: Мир, 1982. 430 с.][М. Саломяк, М. Бирман, Спектральная теория самосопряженных операторов в гильбертовом пространстве. Л.: ЛГУ, 1980. 264 с.][Т. Х. Расулов, Х. Х. Турдиев, “Некоторые спектральные свойства обобщённой модели Фридрихса” // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 2011. № 2(23). С. 181-188. doi: 10.14498/vsgtu904.]