Squares of symmetric operators

Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families $\{{\mathcal{T}}_z{\}}_{\mathrm{I}\mathrm{m}z>0}$ 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 $\mathrm{d}\mathrm{o}\mathrm{m}{\mathcal{T}}_z^2=\{0\}$ has been considered in [5]. Given a densely defined closed symmetric operator $S$, in terms of the intersection of the domain of $S$ with $\mathrm{r}\mathrm{a}\mathrm{n}(S-\lambda I)$ and the projection of the domain of the adjoint $S^{\ast }$ on $\mathrm{r}\mathrm{a}\mathrm{n}(S-\lambda I)$, $\lambda \in \mathbb{C}\setminus \mathbb{R}$, necessary and sufficient conditions for the cases (I)–(III) related to the domain of $S^2$, are obtained.