Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ with dense image.
Define now the group $U(E,H)$ to consist of those invertible, continuous, linear transformations on $E$, which extend to unitary transformations of $H$.
What is known about the group $U(E,H)$?
- Does the map $U(E,H) \rightarrow U(H)$ have dense image?
- Is it a Fréchet Lie group? If so:
- What is its Lie algebra? Is it just the continuous linear skew-adjoint operators on $E$?
- Is the map $U(E,H) \times E \rightarrow E$ smooth?
- Do the smooth vectors of the representation $U(E,H) \times H \rightarrow H$ consist exactly of $E$?
- Is $U(E,H)$ contractible?
I am most interested in the case that $E$ is a nuclear Fréchet space.
Perhaps a good first step towards a Fréchet Lie group structure on $U(E,H)$ would be to equip the space of skew operators on $E$ with the structure of Fréchet (Lie) algebra. It's not so obvious to me how this should go. Is there anything known in this direction?