On complementation of vector-valued Hardy spaces

Abstract

Let X X be a complex Banach space and 1 > p > ∞ 1 > p > \infty . H p ( X ) {H^p}(X) resp. h p ( X ) {h^p}(X) denote the Hardy spaces of X X -valued analytic resp. harmonic functions on the disc. L p ( X ) {L^p}(X) is the Lebesgue-Bochner space of X X -valued integrable functions on the circle and H p ( X ) {{\mathbf {H}}^p}(X) its Hardy-type subspace { f ∈ L p ( X ) : f ^ ( n ) = 0 ∀ n > 0 } \{ 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 ) {H^p}(X) is complemented in h p ( X ) {h^p}(X) ; the canonical analytic (or Riesz) projection is a bounded operator h p ( X ) → H p ( X ) ; H p ( X ) {h^p}(X) \to {H^p}(X);{{\mathbf {H}}^p}(X) is complemented in L p ( X ) {L^p}(X) ; analytic projection is a bounded operator L p ( X ) → H p ( X ) {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 X .