# An operator approach to the indefinite Stieltjes moment problem

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## 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.

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## Author information

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Correspondence to Vladimir Derkach.

Dedicated to E.R. Tsekanovskii on the occasion of his 80th birthday

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 14, No. 1, pp. 42–85, January–March, 2017.

This work was supported by a grant of the Volkswagen Foundation and by the Ministry of Education and Science of Ukraine (project numbers 0115U000556, 0115U000136).

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Derkach, V., Kovalyov, I. An operator approach to the indefinite Stieltjes moment problem. J Math Sci 227, 33–67 (2017). https://doi.org/10.1007/s10958-017-3573-3