An extremal problem for functions annihilated by a Toeplitz operator

Abstract

For a bounded function $\varphi$ on the unit circle ${\mathbb {T}}$, let $T_{\varphi }$ be the associated Toeplitz operator on the Hardy space $H^2$. Assume that the kernel

$$\begin{aligned} K_2(\varphi ):=\left\{ f\in H^2:\,T_{\varphi } f=0\right\} \end{aligned}$$

is nontrivial. Given a unit-norm function f in $K_2(\varphi )$, 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 $f_1,f_2\in K_2(\varphi )$, 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 $\overline{z\varphi f}$ has a nonconstant inner factor. The proof relies on an intrinsic characterization of the moduli of functions in $K_2(\varphi )$, a result which we also extend to $K_p(\varphi )$ (the kernel of $T_{\varphi }$ in $H^p$) with $1\le p\le \infty$.