Abstract
We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.
References
[1] E. Nelson, Analytic vectors, Ann. of Math., 70(3), (1959), 572-615.10.2307/1970331Search in Google Scholar
[2] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, New York (1980).Search in Google Scholar
[3] A.E. Nussbaum, A commutativity theorem for unbounded operators in Hilbert space, Trans. Amer. Math. Soc., vol. 140, (1969) 485-491.Search in Google Scholar
[4] Y. Xu, Unbounded commuting operators and multivariate orthogonal polynomials, Proc. Amer. Math Soc., vol. 119, No 4, (1993) 1223-1231.10.1090/S0002-9939-1993-1165065-7Search in Google Scholar
[5] B.P. Allakhverdiev, G. Sh. Guseinov, On the spectral theory of dissipative difference operators of second order, Math. USSR Sbornik, vol 66, No. 1, (1990), 107-125.10.1070/SM1990v066n01ABEH002081Search in Google Scholar
[6] N.I. Akhiezer, The Classical Moment Problem, Oliver Boyd, Edinburgh (1965).Search in Google Scholar
[7] J. Dombrovski, Cyclic operators, commutators, and absolutely continuous measures, Proc. Amer. Math Soc., vol. 100, No 3, (1987), 457-463.10.1090/S0002-9939-1987-0891145-4Search in Google Scholar
[8] M. Gekhtman, A. Kalyuzhny, On the orthogonal polynomials in several variables, Integr. Equat. Oper. Theory, vol. 19, Isssue 4, (1994), 404-418.10.1007/BF01299841Search in Google Scholar
[9] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications 81, Cambridge University Press, Cambridge (2001).10.1017/CBO9780511565717Search in Google Scholar
[10] V.I. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Kluwer Academic Publishers, Dordrecht (1991).10.1007/978-94-011-3714-0Search in Google Scholar
[11] E.N. Petropoulou, L. Velazquez, Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations, J. Math. Anal. Appl., vol. 420, (2014), 852-872.10.1016/j.jmaa.2014.05.077Search in Google Scholar
[12] Ju. M. Berezanskij, Expansions in Egenfunctions of Selfadjoint Operators, AMS, Providence, R.I. (1968).Search in Google Scholar
[13] E. M. Nikishin, V. N. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs, vol. 92, AMS, Providence, R.I. (1991).Search in Google Scholar
[14] A.G. Kostyuchenko, K.F. Mirzoev, Three-term recurrence relations with matrix coefficients. The completely indeterminate case, Math. Notes, vol. 63, No. 5, (1998) 624-630.10.1007/BF02312843Search in Google Scholar
[15] A. G. Kostyuchenko, K. A. Mirzoev, Complete indefiniteness tests for Jacobi matrices with matrix entries, Functional Analysis and Its Applications, vol.35, No. 4, (2001) 265-269.Search in Google Scholar
[16] A. Osipov, On the completely indeterminate case for block Jacobi matrices. Concrete Operators, vol. 4, No. 1, (2017) 48-57.Search in Google Scholar
[17] B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137, (1998) 82-203.Search in Google Scholar
© 2019 Andrey Osipov, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 Public License.
