TITLE: What is known about the "unitary group" of a rigged Hilbert space?
QUESTION [18 upvotes]: 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 $U(E,H)$?

For example:

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?

REPLY [4 votes]: These are just some thoughts on you question, so not an answer but too long for a comment.  I waited a few days before posting this since, rather than addressing your post directly, I am going to look at a variant.  This is usually not the done thing but my excuse is that I hope that it will contain material which could be of interest to you.
First of all, as has already been pointed out, your questions only make sense if you specify the relevant topologies. Experience suggests that the norm is not suitable for such questions. However, there are available a range of well-behaved and well-studied weaker ones which might do very well.
My main thesis is that most of the rigged Hilbert spaces which arise in mathematical physics have a special form and it is worth while starting there.  The setting is an unbounded self adjoint operator $T$ on a Hilbert space.  $E$ is then the intersection of the domains of definition of its powers.  This is a Fréchet space with its natural structure, even Fréchet nuclear (Pietsch) under conditions on the spectral behaviour of $T$ which are satisfied by many of the classical operators (Laplace, Laplace-Beltrami, Schrödinger).
My main suggestion is that one study the space of unitary operators on the Hilbert space which commute with $T$.  Such operators then map $E$ continuously into itself.
This space can be explicitly calculated for most of the situations mentioned above and this may provide some insight into your original question.
As an example, for the case of the standard one-dimensional Schrödinger operator the Fréchet space is that of the rapidly decreasing smooth functions and the corresponding operator space is the countable product of  circle groups, regarded as multipliers on the coefficients of the Hermite expansion.  As a compact group it won’t be dense anywhere in a non-trivial way but it will, presumably, be an infinite dimensional Lie group.
Similar remarks apply to the other classical cases where the spectral behaviour of the operator is known.