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Every shift invariant subspace of the Hardy space $H^2(\mathbb{D})$ is either $\{0\}$ or is of the form $\varphi H^2$ for some inner function $\varphi$. I know that if $\varphi(0) \neq 0$, then the orthogonal projection of 1 onto $\varphi H^2$ is $\overline{\varphi(0)}\varphi$. Can we say generally what the orthogonal projections of the functions $z^n$ are onto this space?

If this is not possible in general, is it at least possible for the case when $\varphi$ is an infinite Blaschke product not vanishing at the origin?

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  • $\begingroup$ For any subspace $M$ with ONB $\{ e_n\}$, the projection onto $M$ is given by $Px=\sum \langle e_n,x\rangle e_n$. In your case, a natural ONB is $\varphi(z) z^n$, $n\ge 0$. $\endgroup$
    – Christian Remling
    Commented May 12 at 23:38

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