Indeterminate Jacobi operators
Christian Berg, Ryszard Szwarc
TL;DR
The paper analyzes the Jacobi operator $(T,D(T))$ arising from an indeterminate Hamburger moment problem, establishing when linear combinations of the deficiency/second-kind sequences $\mathfrak{p}_u,\mathfrak{p}_v,\mathfrak{q}_u,\mathfrak{q}_v$ belong to $D(T)$ or to the domain of a self-adjoint extension, using four Nevanlinna functions of two variables $(A,B,C,D)$. It proves that $\mathfrak{p}_\lambda$ and $\mathfrak{q}_\lambda$ are never in $D(T)$ and derives explicit criteria: $\mathfrak{p}_u+\alpha\mathfrak{p}_v\in D(T)$ iff $D(u,v)=0$ with $\alpha=B(u,v)$; $\mathfrak{q}_u+\beta\mathfrak{q}_v\in D(T)$ iff $A(u,v)=0$ with $\beta=-C(u,v)$; and $\mathfrak{p}_u+\gamma\mathfrak{q}_v\in D(T)$ iff $B(u,v)=0$ with $\gamma=-D(u,v)$. Self-adjoint extensions $T_t$ are described alongside $N$-extremal spectral measures $\mu_t$, giving criteria for membership of $\mathfrak{p}_\lambda$ and $\mathfrak{q}_\lambda$ in $D(T_t)$. An explicit bounded operator $\Xi_{z_0}$ yields a concrete parametrization of $D(T)$ as its range, connected to a de Branges space of entire functions and a corresponding unitary isomorphism with $\ell^2$. These results provide detailed domain descriptions and spectral parametrizations for Jacobi operators in indeterminate moment problems, enabling boundary-value analysis and spectral reconstruction.
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 $\ell^2$.