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.
Categories: math.FA