On contractive projections in Hardy spaces

We prove a conjecture of Wojtaszczyk that for $1\leq p<\infty$, $p\neq 2$, $H_p(\mathbbT)$ 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.

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