Indeterminate Jacobi operators II

TL;DR

This paper studies the domain of the indeterminate Jacobi operator $(T,D(T))$ arising from a Hamburger moment problem and provides explicit dense subspaces of $\ell^2$ generated from Nevanlinna data. It constructs the dense subspaces $P(v_0)$, $Q(v_0)$, and $M(v_0)$ from $\mathfrak{p}_u$, $\mathfrak{q}_u$, and their linear combinations, and proves their density, with orthogonality relations ensuring optimality. The analysis relies on the Nevanlinna functions $A,B,C,D$ satisfying $A(u)D(u)-B(u)C(u)=1$, the second-kind polynomials $q_n$, and N-extremal measures $\mu_t$ to elucidate when such vectors lie in $D(T)$. Overall, the work refines the understanding of $D(T)$ in the indeterminate setting by providing concrete dense generators and optimality properties applicable to spectral and extension problems.

Abstract

We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, and present countable subsets S of the domain D(T) such that span(S) is dense in \ell^2. As an example we have S={(p_n(u))+B(u)(p_n(0)):D(u)=0}, where (p_n) denotes the orthonormal polynomials of the moment problem and B,D are two of the Nevanlinna functions. It is also proved that sets like S are optimal in the sense that if one vector is removed, then the span is no longer dense.

Theorems & Definitions (9)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• proof
• Lemma 3.2