Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An extremal problem for functions annihilated by a Toeplitz operator   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
$1\le p\le\infty$.

36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
3
The categories of this paper are: physics
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th