On the rank one perturbations of the Heisenberg commutation relation and unbounded subnormal operators (Q1587858)

\({\mathcal A}\) densely defined linear operator \(A\) in a Hilbert space is said to belong to the class \({\mathcal P}\), if there exists a dense linear manifold \(L(A)\subset{\mathcal D}(A)\cap{\mathcal D}(A^*)\) such that the minimal reducing manifold containing \(M(A):= \{[A^*, A]x- x: x\in L(A)\}\) is dense in \(L(A)\). For a subclass of \({\mathcal P}\) two operators forming a complete pair of unitary invariants are established. If \(\dim M(A)= 1\) a functional model for \(A\) is presented, and conditions are given which imply the subnormality of \(A\) to be equivalent to \([A^*, A]\geq I\).