A note on common range of a class of co-analytic Toeplitz operators

Abstract

We characterize the intersection of the ranges of a class of co-analytic Toeplitz operators by considering this set as the dual space of the Privalov space $N^p$, $1<p<\infty$, in a certain topology. For a fixed $p$ we define the class $H_p$ consisting of those de Branges spaces $\mathcal{H}(b)$ such that the function $b$ is not an extreme point of the unit ball of $H^{\infty }$, and the associated measure ${\mu }_b$ for $b$ satisfies an additional condition. It is proved that the function $f$ analytic on $\mathbf{D}$ is a multiplier of every de Branges space from $H_p$ if and only if $f$ is in the intersection of the ranges of all Toeplitz operators belonging to the class $H_p$. We show that this is true if and only if the Taylor coefficients $\hat{f}(n)$ of $f$ decay like $O(\exp (-c{n}^{1/(p+1)}))$ for a positive constant $c$.