A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in the unbounded case (see Squeezing Bogoliubov transformations on the infinite mode CCR-algebra, Honegger and Rieckers, Com. Math. Phys. 37 no. 9 (1996) )
Question: Does a similar result hold for a (not necessarily surjective) symplectic map $T:H\to K$ between Hilbert spaces?
(The reference I have relies pretty heavily on $T$ having dense range)