Paley-Wiener isomorphism over infinite-dimensional unitary groups

An analog of the Paley-Wiener isomorphism for the Hardy space with an invariant measure over infinite-dimensional unitary groups is described. This allows us to investigate on such space the shift and multiplicative groups, as well as, their generators and intertwining operators. We show applications to the Gauss-Weierstrass semigroups and to the Weyl-Schr"odinger irreducible representations of complexified infinite-dimensional Heisenberg groups.