  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 $$K_2(\varphi):=\{f\in H^2:\,T_\varphi f=0\}$$ is
nontrivial. Given a unit-norm function $f$ in $K_2(\varphi)$, we ask whether an
identity of the form $|f|^2=\frac12\left(|f_1|^2+|f_2|^2\right)$ 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 nontrivial 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
$1\le p\le\infty$.
