Jacobi operators with discrete spectrum and growth of the Nevanlinna matrix

Abstract

The Nevanlinna matrix of an indeterminate moment problem is used to parameterize the solutions of the moment problem. Its entries are entire functions of minimal exponential type, and any smaller growth may occur. In theory the growth can be determined from the orthogonal polynomials of the moment sequence, however, the corresponding formula is virtually impossible to apply in practice. We give easy-to-use formulas in another set of parameters associated with the moment sequence by using the connection with canonical systems whose Hamiltonian reflects the discrete nature of the Jacobi operator. Translating the results fully to the Jacobi parameters is very hard, if not impossible. Still, some interesting conclusions can be drawn.