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Journal of Operator Theory

Volume 89, Issue 1, Winter 2023  pp. 87-103.

Selfadjoint Jacobi operators in the limit circle case

Authors:  Dimitri R. Yafaev
Author institution:Universite de Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France and SPGU, Univ. Nab. 7/9, Saint Petersburg, 199034 Russia, and NTU Sirius, Olympiysky av. 1, Sochi, 354340 Russia

Summary: We consider symmetric Jacobi operators with recurrence coefficients such that the corresponding difference equation is in the limit circle case. Equivalently, this means that the associated moment problem is indeterminate. Our main goal is to find a representation for the resolvents of selfadjoint realizations $J$ of such Jacobi operators. This representation implies the classical Nevanlinna formula for the Cauchy--Stieltjes transforms of the spectral measures of the operators $J$. We also efficiently describe domains of the operators $J$ in terms of boundary conditions at infinity.

Keywords: indeterminate moment problems, Jacobi matrices, selfadjoint realizations, resolvents, difference equations, Jost solutions

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