There is possibly a huge literature on the subject but I am a newcomer on analytic representations and my need is rather specific. I simplify it below.
Let $A,B$ be two symbols (standing for annihilation and creation) and set $$ H_\mathbb{C}:=\frac{\mathbb{C}<A,B>}{<[A,B]-1>}=<A,B;[A,B]=1>_{\mathbb{C}-AAU} $$ where $\mathbb{C}<A,B>$ is the free (associative with unity i.e. AAU) algebra and $<[A,B]-1>$ the two sided ideal generated by $[A,B]-1$ (known as Heisenberg or Heisenberg-Weyl - one dimensional - algebra). My question is the following
Are there faithful and handy representations of $H_\mathbb{C}$ which allow the computation of one-parameter groups of its elements ?
The answers can be positive or even non-go theorems (see remarks below).
Remarks : (a) There are lots of variants of Fock spaces to represent faithfully $H_\mathbb{C}$, one of them is the Bargmann-Fock representation ($A=\frac{d}{dz},\ B=z$) which acts on : polynomials, series, entire functions (Hardy space $\mathcal{H}(\mathbb{C})$). The one on entire function allows for computation of the displacement operator as a one-parameter group through $$ e^{t\frac{d}{dz}}[f](z)=f(z+t) $$
the series converges for the topology of compact convergence on $\mathcal{H}(\mathbb{C})$.
(b) An easy non-go theorem is the following :
Theorem There exists no representation of $H_\mathbb{C}$ in a Banach algebra and hence in m-convex Fréchet algebras, the latter being projective limits of the former.
In particular, this proves that, although the space $H_\mathbb{C}=C^\omega(\mathbb{C},\mathbb{C})$ (the space of complex entire functions) can be endowed with the structure of a m-convex Fréchet algebra (the standard topology of compact convergence), the algebra of its continuous operators $\mathcal{L}(C^\omega(\mathbb{C},\mathbb{C}))$ admits no such structure.
