Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)2886410.22363/2413-3639-2021-67-2-237-254ArticlesСтатьиResearch ArticleDeficiency Indices of Block Jacobi Matrices: SurveyИндексы дефекта блочных матриц Якоби: обзорBudykaViktoriya S.БудыкаВиктория Сергеевнаbudyka.vik@gmail.comMalamudMark M.МаламудМарк Михайловичmalamud3m@gmail.comMirzoevKarahan A.МирзоевКарахан Агахан оглыmirzoev.karahan@mail.ruPeoples Friendship University of Russia (RUDN University)Российский университет дружбы народовDonetsk Academy of Management and Public AdministrationДонецкая академия управления и государственной службыLomonosov Moscow State UniversityМосковский государственный университет имени М. В. ЛомоносоваMoscow Center for Fundamental and Applied MathematicsМосковский центр фундаментальной и прикладной математики15122021672Dedicated to the memory of Professor N. D. KopachevskyПосвящается памяти профессора Н. Д. Копачевского23725423102021Copyright ©; 2021, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2021, Современная математика. Фундаментальные направления2021Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.enhttps://journals.rudn.ru/CMFD/article/view/28864

The paper is a survey and concerns with infinite symmetric block Jacobi matrices J with m×m-matrix entries. We discuss several results on general block Jacobi matrices to be either self-adjoint or have maximal as well as intermediate deficiency indices. We also discuss several conditions for J to have discrete spectrum.

Работа является обзорной. Ее основной объект - бесконечные симметричные блочные матрицы Якоби J с m×m-матричными элементами. Обсуждаются результаты, в которых общие блочные матрицы Якоби являются самосопряженными или могут иметь максимальные либо промежуточные индексы дефекта. Также обсуждаются условия, гарантирующие дискретность спектра матриц Якоби J.

Коган В. И. Об операторах, порожденных Ip-матрицами, в случае максимальных индексов дефекта// Теор. функций, функц. анализ. и их прилож. - 1970. - 11. - C. 103-107Крейн М. Г. Бесконечные J -матрицы и матричная проблема моментов// Докл. АН СССР. - 1949. - 69, № 2. - C. 125-128.Крейн М. Г. Основные положения теории представления эрмитовых операторов с индексом дефекта (m, m)// Укр. мат. ж. - 1949. - 1, № 2. - C. 3-66.Akhiezer N. I. The classical moment problem and some related questions in analysis. - Edinburgh-London: Oliver & Boyd Ltd, 1965.Berezansky Ju. M. Expansions in eigenfunctions of self-adjoint operators. - Providence: AMS, 1968.Braeutigam I. N., Mirzoev K. A. Deficiency numbers of operators generated by infinite Jacobi matrices// Dokl. Math. - 2016. - 93, № 2. - C. 170-174.Braeutigam I. N., Mirzoev K. A. On deficiency numbers of operators generated by Jacobi matrices with operator elements// St. Petersburg Math. J. - 2019. - 30, № 4. - C. 621-638.Budyka V. S., Malamud M. M. On the deficiency indices of block Jacobi matrices related to Dirac operators with point interactions// Math. Notes. - 2019. - 106. - C. 1009-1014.Budyka V. S., Malamud M. M. Self-adjointness and discreteness of the spectrum of block Jacobi matrices// Math. Notes. - 2020. - 108. - C. 445-450.Budyka V. S., Malamud M. M. Deficiency indices of Jacobi matrices and Dirac operators with point interactions on a discrete set// ArXiv. - 2021. - 2012.15578.Budyka V. S., Malamud M. M., Posilicano A. To spectral theory of one-dimensional matrix Dirac operators with point matrix interactions// Dokl. Math. - 2018. - 97. - C. 115-121.Carlone R., Malamud M., Posilicano A. On the spectral theory of Gesztesy-Sˇ eba realizations of 1-D Dirac operators with point interactions on a discrete set// J. Differ. Equ. - 2013. - 254, № 9. - C. 3835-3902.Chihara T. Chain sequences and orthogonal polynomials// Trans. Am. Math. Soc. - 1962. - 104.- C. 1- 16.Cojuhari P., Janas J. Discreteness of the spectrum for some unbounded matrices// Acta Sci. Math. - 2007. - 73. - C. 649-667.Dombrowski J., Pedersen S. Orthogonal polynomials, spectral measures, and absolute continuity// J. Comput. Appl. Math. - 1995. - 65. - C. 115-124.Dyukarev Yu. M. Deficiency numbers of symmetric operators generated by block Jacobi matrices// Sb. Math. - 2006. - 197, № 8. - C. 1177-1203.Dyukarev Yu. M. Examples of block Jacobi matrices generating symmetric operators with arbitrary possible values of the deficiency numbers// Sb. Math. - 2010. - 201, № 12. - C. 1791-1800.Dyukarev Yu. M. On conditions of complete indeterminacy for the matricial hamburger moment problem// В сб.: «Complex Function Theory, Operator Theory, Schur Analysis and Systems Theory». - Cham: Birkha¨user, 2020. - С. 327-353.Janas J., Naboko S. Multithreshold spectral phase transition for a class of Jacobi matrices// Oper. Theory Adv. Appl.- 2001.- 124. - C. 267-285.Kostenko A. S., Malamud M. M. One-dimensional Schro¨ dinger operator with δ-interactions// Funct. Anal. Appl.- 2010.- 44, № 2. - C. 151-155.Kostenko A. S., Malamud M. M. 1-D Schro¨ dinger operators with local point interactions on a discrete set// J. Differ. Equ. - 2010. - 249, № 2. - C. 253-304.Kostenko A. S., Malamud M. M. 1-D Schro¨ odinger operators with local point interactions: a review// Proc. Sympos. Pure Math. - 2013. - 87. - C. 232-262.Kostenko A. S., Malamud M. M., Natyagailo D. D. Matrix Schro¨ dinger operator with δ-interactions// Math. Notes. - 2016. - 100, № 1. - C. 49-65.Kostyuchenko A. G., Mirzoev K. A. Three-term recurrence relations with matrix coefficients. The completely indefinite case// Math. Notes. - 1998. - 63, № 5-6. - C. 624-630.Kostyuchenko A. G., Mirzoev K. A. Generalized Jacobi matrices and deficiency numbers of ordinary differential operators with polynomial coefficients// Funct. Anal. Appl. - 1999. - 33. - C. 25-37.Kostyuchenko A. G., Mirzoev K. A. Complete indefiniteness tests for Jacobi matrices with matrix entries// Funct. Anal. Appl.- 2001.- 35. - C. 265-269.Malamud M. M. On a formula of the generalized resolvents of a nondensely defined Hermitian operator// Ukr. Math. J. - 1992. - 44. - C. 1522-1547.Malamud M. M., Malamud S. M. Spectral theory of operator measures in Hilbert space// St. Petersbg. Math. J.- 2004.- 15, № 3. - C. 323-373.Mirzoev K. A., Konechnaya N. N., Safonova T. A., Tagirova R. N. Generalized Jacobi matrices and spectral analysis of differential operators with polynomial coefficients// J. Math. Sci. (N.Y.) - 2021. - 252, № 2. - C. 213-224.Mirzoev K. A., Safonova T. A. On the deficiency index of the vector-valued Sturm-Liouville operator// Math. Notes. - 2016. - 99, № 2. - C. 290-303.Petropoulou E., Vela´ zquez L. Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations// J. Math. Anal. Appl. - 2014. - 420. - C. 852-872.S´ widerski G. Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime// J. Approx. Theory. - 2018. - 233.- C. 1-36S´ widerski G. Spectral properties of block Jacobi matrices// Constr. Approx. - 2018. - 48, № 2. - C. 301- 335