DEFICIENCY INDICES OF THE OPERATORS GENERATED BY INFINITE JACOBI MATRICES WITH OPERATOR ENTRIES

摘要

Let J be an infinite symmetric Jacobi matrix whose entries are either linear operators acting in the finite dimensional space C-m or bounded linear operators acting on an infinite-dimensional separable Hilbert space H. The minimal closed symmetric operator L induced by J is considered in the Hilbert spaces l(2)(N-0, C-m) or l(2)(N-0, H), respectively. New criteria are given for the minimality, maximality, and nonmaximality of the deficiency indices of this operator, i.e., criteria in terms of the matrix J for the corresponding moment problem to be determinate, completely indeterminate and noncompletely indeterminate. The main emphasis is on conditions on the entries of a numerical Jacobi matrix that ensure the determinate or indeterminate cases of the classical moment problem. These results are applied to a construction of examples of entire operators (in the sense of M. Krein) with infinite deficiency indices as well as to the Sturm-Liouville vector differential operator with point interactions on the semiaxis.