An operator approach to the indefinite Stieltjes moment problem

Abstract

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N_κ (κ ϵ ℤ₊), if f is symmetric with respect to ℝ and the kernel \( {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} \) has κ negative squares on ℂ₊. The generalized Stieltjes class \( {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) \) is defined as the set of functions f ϵ N_κ such that z f ϵ N_k. The full indefinite Stieltjes moment problem \( {MP}_{\kappa}^k\left(\mathbf{s}\right) \) consists in the following: Given κ, k ϵ ℤ₊, and a sequence \( \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } \) of real numbers, to describe the set of functions \( f\in {\mathbf{N}}_{\kappa}^k \), which satisfy the asymptotic expansion

\( f(z)=-\frac{s_0}{z}-\cdots -\frac{s_2n}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right)\kern1em \left(z=-y\in {\mathrm{\mathbb{R}}}_{-},y\uparrow \infty \right) \)

for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem \( {MP}_{\kappa}^k\left(\mathbf{s}\right) \) within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A_[0;N] generated by \( {\mathfrak{J}}_{\left[0;N\right]} \). The u-resolvent matrices of the operator A_[0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem \( {MP}_{\kappa}^k\left(\mathbf{s}\right) \) to be solvable and indeterminate are found. Explicit formulae for Padé approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.