(joint with Malte Gerhold)

So-called Interacting Fock spaces have been introduced in the nineties by Accardi, Lu, and Volovich as a far-reaching generalization of Fock spaces: An $N_0$-graded Hilbert space \[ {\mathcal I}=\bigoplus_{k=0}^n H_n \]
with with one-dimensional degree-zero space $H_0=\Omega C$ and a set $\mathcal A$ of degree-one operators that ``create'' everything out of the vacuum unit vector $\Omega$:
\[ span {\mathcal A}H_n = H_{n+1}. \]
In this form, the definition is from our recent joint work with Gerhold.

Not only for applications it is of outstanding importance that the definition is actually for pre-Hilbert spaces (direct sum and linear span are algebraic). But also if we were interested only in the non-selfadjoint operator algebras generated by a set $\mathcal A$ of bounded creators, we definitely would loose examples if we insisted to model them on Hilbert spaces, only.

Curious? Well, of course, if we have as set of bounded operators, there is no problem to complete the domains. The secret is in the parametrization of interacting Fock spaces. We have an efficient parametrization of all interacting Fock spaces (including the set $\mathcal A$) by certain degree-zero operators. (This has similarity with things that hapen with weighted shifts.) It is no surprise that these operators have to be allowed to be unbounded, when we are characterizing all interacting Fock spaces. However, it turns out that even interacting Fock spaces having only bounded creators may have unbounded operators characterizing them.