Generalized selfadjointness of operators generated by Jacobi Hermitian matrices

Abstract

We investigate selfadjointness in sense of Hilbert space rigging and related questions. We proved that this generalized selfadjointness of some operator, which acts from positive into negative space, is equivalent to ordinary selfadjointness of some modification of this operator in basic ("zero") space.

Also we consider operators generated by classical and generalized Jacobi Hermitian matrices, their selfadjointness and generalized selfadjointness in sense of weight Hilbert space rigging. Some sufficient conditions of generalized selfadjointness of these operators are proved. Using obtained results we explaine possibility of construction of example of gereralized selfadjoint opearator which is not selfadjoint in classical sence.