Universität Osnabrück, Germany
i.m.kovalyov@gmail.com
Indefinite Stieltjes Moment Problem and Jacobi Matrices
Friday, 29 September 2023 at 13:00
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
Let \(\mathbf{J}\) be a monic generalized Jacobi matrix, i.e. a three-diagonal block matrix of a special form, introduced by M. Derevyagin and V. Derkach in 2004. We find conditions for a monic generalized Jacobi matrix \(\mathbf{J}\) to admit a factorization \(\mathbf{J} = \mathbf{L}\mathbf{U}\) with \(\mathbf{L}\) and \(\mathbf{U}\) being lower and upper triangular two-diagonal block matrices of a special form. In this case the Darboux transformation of \(\mathbf{J}\) defined by \(\mathbf{J}^p = \mathbf{L}\mathbf{U}\) it is shown to be also a monic generalized Jacobi matrix. Analogues of Christoffel formulas for polynomials of the first and the second kind, corresponding to the Darboux transformation \(\mathbf{J}^p\) are found. As was known, the generalized Jacobi matrix is associated with the indefinite Stieltjes moment problem. Hence, we consider one in the class \(N_{\kappa}^k\) of generalized Stieltjes functions. The set of the solution can be described by Schur step-by-step algorithm, which is based on the expansion of the solutions in the generalized Stieltjes continued fraction. The resolvent matrix is represented in terms of generalized Stieltjes polynomials.
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