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 < ∞ , in a certain topology. For a fixed p we define the class H_p consisting of those de Branges spaces \mathscr{H}(b) such that the function b is not an extreme point of the unit ball of H^∞ , and the associated measure μ_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 (−cn^{1/(p + 1)})) for a positive constant c .