Causal compactification and Hardy spaces for spaces of Hermitian type

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Let G/H be a compactly causal symmetric space with causal compactification Φ : G/H → Š1, where Š1 is the Bergman-Šilov boundary of a tube type domain G1/K1. The Hardy space H2(C) of G/H is the space of holomorphic functions on a domain Ξ(C°) ⊂ Gℂ/Hℂ with L2-boundary values on G/H. We extend Φ to imbed Ξ(C°) into G1/K1, such that Ξ(C°) = {z ∈ G1/K1 | ψm(z) ≠ 0}, with ψm explicitly known. We use this to construct an isometry I of the classical Hardy space Hcl on G1/K1 into H2(C) or into a Hardy space H̃2(C) defined on a covering Ξ̃(C°) of Ξ(C°). We describe the image of I in terms of the highest weight modulus occuring in the decomposition of the Hardy space.

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Pacific Journal of Mathematics

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