On a criterion for the determinate-indeterminate dichotomy of the moment problem

When the classical Hamburger moment problem has solutions, it has either exactly one solution or infinitely many solutions. Correspondingly, the moment problem is said to be either determinate or indeterminate. In terms of Jacobi operators, this dichotomy translates into the operator being either selfadjoint or symmetric nonselfadjoint. In this work, we present a new criterion for the determinate-indeterminate classification which hinges on bases of representation (in Akhiezer-Glazman terminology) for Jacobi operators so that the corresponding matrices have a certain structure.