An extremal problem for functions annihilated by a Toeplitz operator

Abstract

For a bounded function $\phi$ on the unit circle $\mathbb{T}$, let $T_{\phi }$ be the associated Toeplitz operator on the Hardy space $H^2$. Assume that the kernel $$K_2(\phi ):=\{f\in H^2:T_{\phi }f=0\}$$ is nontrivial. Given a unit-norm function $f$ in $K_2(\phi )$, we ask whether an identity of the form $|f{|}^2=\frac{1}{2}\left(|f_1{|}^2+|f_2{|}^2\right)$ may hold a.e. on $\mathbb{T}$ for some $f_1,f_2\in K_2(\phi )$, both of norm $1$ and such that $|f_1|\ne |f_2|$ on a set of positive measure. We then show that such a decomposition is possible if and only if either $f$ or $\overline{z\phi f}$ has a nontrivial inner factor. The proof relies on an intrinsic characterization of the moduli of functions in $K_2(\phi )$, a result which we also extend to $K_p(\phi )$ (the kernel of $T_{\phi }$ in $H^p$) with $1\le p\le \infty$.

10 pages

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