On complementation of vector-valued Hardy spaces

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 $X$.