On contractive projections in Hardy spaces

Florence Lancien, Beata Randrianantoanina, Eric Ricard
2005 Studia Mathematica  
We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p = 2, H p (T) does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p = 2, H p does not admit a Schauder basis with constant one. 2000 Mathematics Subject Classification: 46E15, 30D55, 46B20, 46B04. B. Randrianantoanina was a participant in NSF Workshop in Linear Analysis and Probability, Texas A&M University. [93] Preliminaries Definition 2.1. Let X
more » ... inition 2.1. Let X be a Banach space. We define the duality map J from X into subsets of X * by the condition that f ∈ J(x) ⊂ X * if and only if f X * = x X and f, x = x 2 X . We note that if X is a strictly convex Banach space, then for all x ∈ X, the set J(x) contains exactly one functional. In this case we will consider J as a map from X to X * . If, in addition, X is reflexive and X * is strictly convex, then J * : X * → X, and J * = J −1 . Calvert [3] proved an important characterization of contractively complemented subspaces of reflexive Banach spaces in terms of the duality map J.
doi:10.4064/sm171-1-5 fatcat:wm6gfvr6bbd4tbpzv7aenutejy