On One Class of Vector Continued Fractions with Operator Elements and the Jacobi–Perron Algorithm

Abstract

We consider a class of infinite vector continued fractions of a complex variable such that their coefficients are bounded operators in a Hilbert space. They may be regarded as analogs (in a broad sense) of \(J\)-fractions used in the theory of Jacobi operators and the classical moment problem. To each of the continued fractions under consideration, there corresponds the band operator generated by certain infinite block matrix containing a finite number of nonzero diagonals, which are composed of the (operator) elements of this continued fraction. Using the inverse spectral theory for these band operators, we establish the main properties of such continued fractions, in particular, their expansion algorithm and the criterion for existence of this expansion. It turns out that the algorithm of reconstruction of a band operator from its spectral data (the moment sequence of its Weyl function) can be regarded as a modified version of a known Jacobi-Perron expansion algorithm, applied to a system of operator-functions holomorphic at infinity in order to get a continued fraction from the class under study. Certain issues of the theory of Hermite-Padé approximants, related to the studied subject, are also considered.

DOI 10.1134/S1061920825030148