Indeterminate Jacobi operators

TL;DR

This paper investigates the domain structure of indeterminate Jacobi operators linked to Hamburger moment problems, characterizing self-adjoint extensions using Nevanlinna functions and explicit bounded operators.

Contribution

It provides a detailed analysis of the domains of Jacobi operators in indeterminate cases, connecting them with Nevanlinna functions and constructing explicit bounded operators.

Findings

Criteria for linear combinations of orthogonal polynomials to belong to the domain.

Characterization of self-adjoint extensions via Nevanlinna functions.

Explicit construction of bounded operators with common domain.

Abstract

We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, i.e., the operator in $\ell^2$ defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely many non-zero terms. It is well-known that it is symmetric with deficiency indices (1,1). For a complex number z let $\mathfrak{p}_z, \mathfrak{q}_z$ denote the square summable sequences (p_n(z)) and (q_n(z)) corresponding to the orthonormal polynomials p_n and polynomials q_n of the second kind. We determine whether linear combinations of $\mathfrak{p}_u,\mathfrak{p}_v,\mathfrak{q}_u,\mathfrak{q}_v$ for complex u,v belong to D(T) or to the domain of the self-adjoint extensions of T in $\ell^2$. The results depend on the four Nevanlinna functions of two variables associated with the moment problem. We also show that D(T) is the common range of an explicitly constructed family of bounded operators on $\ell^2$.