An operator approach to the indefinite Stieltjes moment problem (Q1683339)

A meromorphic function \(f\) in \(\mathbb{C}\backslash\mathbb{R}\), which is symmetric with respect to the real line, belongs to the generalized Nevanlinna class \(N_k\) if the kernel \(N(z,w)=(z-\bar w)^{-1}(f(z)-\overline{f(w)})\) has \(k\) negative squares in \(\mathbb{C}_+\), and to the generalized Stieltjes class \(N_k^l\) if \(f\in N_k\) and \(zf\in N_l\). The authors apply an operator-theoretic approach for solving truncated and full indefinite moment problems, that is, the problems, given a sequence of real numbers \(S=\{s_j\}_{j=0}^n\), \(n\leq\infty\), to describe the set of functions \(f\in N_k^l\) which satisfy the following asymptotic expansion \[ f(z)=-\frac{s_0}{z}-\frac{s_1}{z^2}-\cdots\frac{s_n}{z^{n+1}}+o\Bigl(\frac1{z^{n+1}}\Bigr), \] as \(z\) tends to infinity in a nontangential way. To this end they introduce a symmetric linear operator with deficiency indices \((1,1)\) which acts in a certain indefinite inner product space. Based on the theory of boundary triples and Krein's theory of the resolvent matrices, the authors explicitly construct the \(u\)-resolvent matrices of this operator, which provide the solution of the above moment problems.