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 $<\psi, \langle \psi, A \psi>$. psi \rangle$. Furthermore one recognize, recognizes that each Killing-vector Killing vector field on $PH$ can be constructed in this way.
I am interested in the case on of an unbounded, densely defined operator $A$. By Chernoff/Marsden "Properties of infinite dimensional Hamiltonian systems" I know that the Hamilton-vector Hamilton vector field $X$ for $<\psi, \langle \psi, A \psi>, psi \rangle, \psi \in D(A)$ is only defined densely, but generates a globally global flow. It should also leave the metric invariant. Is the converse also analogues analogous to the unbounded bounded case, i. ei.e. are all densely defined Killing-vector Killing vector fields on $PH$ are associated to an unbounded , self-adjoint operator on $H$?
Thanks and a happy new year! Tobias