Squares of symmetric operators

Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families \ Tz_ Im\, z>0 of closed densely defined symmetric operators with the properties: (I) the domain of Tz² is a core of Tz, (II) the domain of Tz² is dense but note a core of Tz, (III) the domain of Tz² is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case dom\, Tz²=\0\ has been considered in [5]. Given a densely defined closed symmetric operator S, in terms of the intersection of the domain of S with ran\, (S-λ I) and the projection of the domain of the adjoint S^* on ran\, (S-λ I), λ∈ C R, necessary and sufficient conditions for the cases (I)--(III) related to the domain of S², are obtained.