Extension of Norm One Projections

Abstract: Through the establishment of several extension theorems, we provide explicit expressions for norm one 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 1-complemented subspaces of $H^p(\mathbb{T})$ with dimension greater than one are isometric to $H^p(\mathbb{T})$; second, all norm one projections on $H^p(\mathbb{T})$ are restrictions of norm one projections on $L^p(\mathbb{T})$ that leave $H^p(\mathbb{T})$ invariant. The second corollary answers a problem posed by P. Wojtaszczyk in 2003.