Explicit Bargmann-type isomorphism between Berezin and Smolyanov representations of bosonic Fock spaces

We construct a Bargmann-type isomorphism defined by the one-particle part $H$ of the Fock space $\Gamma(H)$ for an infinite-dimensional space $H$ with involution. The formulas obtained also make sense in the case $\dim H<\infty$ and are closely related to the Segal–Bargmann space. Central to the construction is the notion of a shift-invariant distribution in the case of an infinite-dimensional domain of test functions.