V. A. Derkach

Full indefinite Stieltjes moment problem is studied via the step-by-step Schur algorithm. Naturally associated with indefinite Stieltjes moment problem are generalized Stieltjes continued fraction and a system of difference equations, which, in turn, lead to factorization of resolvent matrices of indefinite Stieltjes moment problem. A criterion for such a problem to be indeterminate in terms of continued fraction is found and a complete description of its solutions is given in the indeterminate case. Explicit formulas for diagonal and sub-diagonal Padé approximants for formal power series corresponding to indefinite Stieltjes moment problem and convergence results for Padé approximants are presented.

Let $J$ be an $m\times m$ signature matrix, i.e., $J=J^*=J^{-1}$. An $m\times m$ mvf (matrix valued function) $W(\lambda)$ that is meromorphic in the unit disk $\mathbb{D}$ is called $J$-inner if $W(\lambda)JW(\lambda)^*\leq J$ for every $\lambda$ from $\mathfrak{h}_W^+$, the domain of holomorphy of $W$, in ${\mathbb{D}}$, and $W(\mu)JW(\mu)^*= J$ for a.e. $\mu\in\mathbb{T}=\partial \mathbb{D}$. A $J$-inner mvf $W(\lambda)$ is called $A$-singular if it is outer and it is called right $A$-regular if it has no non-constant $A$-singular right divisors. As was shown by D. Arov [18] every $J$-inner mvf admits an essentially unique $A$-regular--$A$-singular factorization $W=W^{(1)}W^{(2)}$. In the present paper this factorization result is extended to the class ${\mathcal U}_\kappa^r(J)$ of right generalized $J$-inner mvf's introduced in~\cite{DD09}. The notion and criterion of $A$-regularity for right generalized $J$-inner mvf's are presented. The main result of the paper is that we find a criterion for existence of an $A$-regular--$A$-singular factorization for a rational generalized $J$-inner mvf.

With each sequence of real numbers ${\mathbf s}=\{s_j\}_{j=0}^\infty$ two kinds of continued fractions are associated, - the so-called $P-$fraction and a generalized Stieltjes fraction that, in the case when ${\mathbf s}=\{s_j\}_{j=0}^\infty$ is a sequence of moments of a probability measure on $\mathbb R_+$, coincide with the $J-$fraction and the Stieltjes fraction, respectively. A subclass $\mathcal H^{reg}$ of regular sequences is specified for which explicit formulas connecting these two continued fractions are found. For ${\mathbf s}\in\mathcal H^{reg}$ the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.