Extension of contractive projections

TL;DR

This paper characterizes all contractive projections and 1-complemented subspaces in Hardy spaces $H^p( ext{T})$ for $p eq 2$, providing explicit formulas and answering a longstanding question in Banach space theory.

Contribution

It provides explicit expressions for contractive projections in Hardy spaces and characterizes 1-complemented subspaces, extending the understanding of these structures.

Findings

All nontrivial 1-complemented subspaces are isometric to $H^p( ext{T})$.

All contractive projections are restrictions of those on $L^p( ext{T})$ that leave $H^p( ext{T})$ invariant.

The results answer a question posed by P. Wojtaszczyk in 2003.

Abstract

Through the establishment of several extension theorems, we provide explicit expressions for all contractive projections and 1-complemented subspaces in the Hardy space $H^p(\mathbb{T})$ for $1\leq p<\infty$, $p\neq 2$. Our characterization leads to two corollaries: first, all nontrivial 1-complemented subspaces of $H^p(\mathbb{T})$ are isometric to $H^p(\mathbb{T})$; second, all contractive projections on $H^p(\mathbb{T})$ are restrictions of contractive projections on $L^p(\mathbb{T})$ that leave $H^p(\mathbb{T})$ invariant. The first corollary provides examples of prime Banach spaces \emph{in the isometric sense}, while the second answers a question posed by P. Wojtaszczyk in 2003.