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On complementation of vector-valued Hardy spaces

Author: Wolfgang Hensgen
Journal: Proc. Amer. Math. Soc. 104 (1988), 1153-1162
MSC: Primary 46E40; Secondary 30D55, 42B30, 46J15
MathSciNet review: 933514
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Abstract: Let $ X$ be a complex Banach space and $ 1 < p < \infty $. $ {H^p}(X)$ resp. $ {h^p}(X)$ denote the Hardy spaces of $ X$-valued analytic resp. harmonic functions on the disc. $ {L^p}(X)$ is the Lebesgue-Bochner space of $ X$-valued integrable functions on the circle and $ {{\mathbf{H}}^p}(X)$ its Hardy-type subspace $ \{ f \in {L^p}(X):\hat f(n) = 0\forall n < 0\} $. It is proved that the following four conditions are equivalent: $ {H^p}(X)$ is complemented in $ {h^p}(X)$; the canonical analytic (or Riesz) projection is a bounded operator $ {h^p}(X) \to {H^p}(X);{{\mathbf{H}}^p}(X)$ is complemented in $ {L^p}(X)$; analytic projection is a bounded operator $ {L^p}(X) \to {{\mathbf{H}}^p}(X)$. It is well known that the last condition, in turn, is equivalent to the UMD property of $ X$.

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Keywords: Vector-valued Hardy spaces, analytic (or Riesz) projection, UMD Banach spaces
Article copyright: © Copyright 1988 American Mathematical Society