On contractive projections in Hardy spaces

We prove a conjecture of Wojtaszczyk that for $1\le p<\mathrm{\infty }$, $p\mathrm{\ne }2$, $H_p(\text{mathbbT})$ does not admit any norm one projections with dimension of the range finite and bigger than 1. This implies in particular that for $1\leq p<\infty$,$p\ne 2$,$H_p$ does not admit a Schauder basis with constant one.