the following is well-known (e. g. Ashtekar/Schilling, Brody/Hughston): A bounded self-adjoint operator $A$ on a Hilbert space $H$ induces a globally defined vector field $X$ on the projective Hilbert space $PH$, which is a symmetry of the Fubini-Study metric. $X$ is the Hamilton-vector field for the function $\langle \psi, A \psi \rangle$. Furthermore one recognizes that each Killing vector field on $PH$ can be constructed in this way.
I am interested in the case of an unbounded, densely defined operator $A$. By Chernoff/Marsden "Properties of infinite dimensional Hamiltonian systems" I know that the Hamilton vector field $X$ for $\langle \psi, A \psi \rangle, \psi \in D(A)$ is only defined densely, but generates a global flow. It should also leave the metric invariant. Is the converse also analogous to the bounded case, i.e. are all densely defined Killing vector fields on $PH$ associated to an unbounded self-adjoint operator on $H$?
Thanks and a happy new year! Tobias