Radial boundary values of Poisson integrals on infinite-dimensional balls

Abstract

We consider a Gelfand triple $E'\rightarrow H\rightarrow E$ , so that E is a separable complex Banach space with dual $E'$ , and H is its dense Hilbert subspace. We investigate the problem of analytic extensions on an open ball $\mathcal{Q}\subset E'$ and their radial boundary values in the Hardy spaces $\mathcal{H}_{\mu}^{p}$ ( $1\le p\le\infty$ ) using the Poisson integrals on the unitary group $U(\infty)$ over H endowed with an invariant probability measure μ. For this purpose, we construct a Poisson-type kernel with the help of the symmetric Fock space Γ generated by H and prove that the set of radial boundary values of these analytic functions entirely coincides with $\mathcal{H}_{\mu}^{p}$ .