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F@Bergman and Hardy spaces for ${\cal DFN}$-domains via holomorphic liftings

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As a model space for quantum mechanical operators I.E. Segal and V. Bargmann invented the Hilbert space $H$ of Gaussian square integrable entire functions on $C^n$. In the following the class of Toeplitz operators on H had attracted the interest of many authors. By replacing $C^n$ with some complex Hilbert space $V$ equipped with a Gaussian measures these notions where generalized to infinite dimensional domains by J. Janas and K. Rudol. New phenomena arise, having no counterpart in the finite dimensional setting and seem to be related to the topological properties of the holomorphic functions on $V$ . In this talk we define Bergman and Hardy spaces for open subsets in a class of i.g. infinite dimensional nuclear spaces. To be more precise: Let $E$ be the dual of a Fr\'{e}chet nuclear space (${\cal DFN}$-space). Then it is well-known that for each open set $U subset E$ the space ${\cal H}(U)$ of all holomorphic functions on $U$ is a nuclear Fr\'{e}chet space. This topological property can be used to construct Bergman spaces of holomorphic functions on $U$. In the case where $E$ is a nuclear inductive limit of Hilbert spaces we derive estimates for the Bergman kernel $K_U$ restricted to the diagonal. Let ${\cal A}$ be a unital Banach-subalgebra of the bounded holomorphic functions on $U$ which separates points. Applying the nuclearity of ${\cal H}(U)$ we show that the evaluation on $U$ is given by an integral formula over the Shilov boundary of ${\cal A}$. Via holomorphic liftings we obtain Szeg\"{o}-kernels together with some boundary estimates. Finally we show that there also is a notion of a Hardy spaces for finite and infinite dimensional domains with arbitrary boundary.

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