Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families $\{ { \mathcal T}_z\}_{ { \rm Im\,} z>0}$ of closed densely defined symmetric operators with the properties: (I) the domain of ${ \mathcal T}_z^2$ is a core of ${ \mathcal T}_z$, (II) the domain of ${ \mathcal T}_z^2$ is dense but note a core of ${ \mathcal T}_z$, (III) the domain of ${ \mathcal T}_z^2$ is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case ${ \rm dom\,} { \mathcal T}_z^2=\{ 0\}$ has been considered in [5]. Given a densely defined closed symmetric operator $S$, in terms of the intersection of the domain of $S$ with ${ \rm ran\,} (S-\lambda I)$ and the projection of the domain of the adjoint $S^*$ on ${ \rm ran\,} (S-\lambda I)$, $\lambda\in{ \mathbb C}\setminus{ \mathbb R}$, necessary and sufficient conditions for the cases (I)--(III) related to the domain of $S^2$, are obtained.
Categories: math.FA