Deficiency indices of the operators generated by infinite Jacobi matrices with operator entries

Abstract:

Let $\mathbf {J}$ be an infinite symmetric Jacobi matrix whose entries are either linear operators acting in the finite dimensional space $\mathbb {C}^m$ or bounded linear operators acting on an infinite-dimensional separable Hilbert space $\mathrm {H}$. The minimal closed symmetric operator $L$ induced by $\mathbf {J}$ is considered in the Hilbert spaces $\ell ^2(\mathbb {N}_0,\mathbb {C}^m)$ or $\ell ^2(\mathbb {N}_0, \mathrm {H})$, 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 $\mathbf {J}$ 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. Kreĭn) with infinite deficiency indices as well as to the Sturm–Liouville vector differential operator with point interactions on the semiaxis.