On Some Properties and Applications of Operator Continued $$J$$-Fractions

We consider a certain class of infinite continued fractions such that their elements are bounded operators in a Hilbert space. They can be regarded as analogs of $$J$$ -fractions related to the classical moment problem and the theory of Jacobi operators. To each of these operator $$J$$ -fractions there corresponds a band operator generated by three-diagonal infinite matrix which entries coincide with the elements of this continued fraction. Using the theory of such band operators, we establish the basic properties of the continued fractions under consideration: their expansion algorithm, a criterion for existence of this expansion, and the uniqueness theorem. Also we establish the convergence (at a geometric rate) of an operator $$J$$ -fraction outside the numerical range of the corresponding band operator to the Weyl function of the latter. We show how these results can be applied for solving quadratic operator equations. DOI 10.1134/S1061920824040125