Bounded symbols of Toeplitz operators on Paley-Wiener spaces and a weak factorization theorem

Abstract

A classical result by R. Rochberg says that every bounded Toeplitz operator $T$ on the Hilbert Paley-Wiener space $\mathrm{PW}_a^2$ admits a bounded symbol $\varphi$. We generalize this result to Toeplitz operators on the Banach Paley-Wiener spaces $\mathrm{PW}_a^p$, $1<p<+\infty$. The Toeplitz commutator theorem describes the integral identity that must hold for a bounded operator $T$ on $\mathrm{PW}_a^p$ to be a Toeplitz operator on $\mathrm{PW}_a^p$. We prove this theorem in the continuous case, thus extending the result previously obtained by D. Sarason in the discrete case. Upon combining the results, we establish the weak factorization theorem, namely, for $p,q>1$, $\frac{1}{p}+\frac{1}{q}=1$, any function $h$ belonging to $\mathrm{PW}^1_{2a}$ can be represented as $$h=\sum_{k\geqslant 0}f_k\bar{g}_k,\qquad f_k\in\mathrm{PW}_a^p,\,g_k\in\mathrm{PW}_a^q.$$