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 Np, 1 < p < ∞, in a certain topology. For a fixed p we define the class Hp consisting of those de Branges spaces ℋ(b) such that the function b is not an extreme point of the unit ball of H∞, and the associated measure μb for b satisfies an additional condition. It is proved that the function f analytic on D is a multiplier of every de Branges space from Hp if and only if f is in the intersection of the ranges of all Toeplitz operators belonging to the class Hp. We show that this is true if and only if the Taylor coefficients \hat{f}(n) of f decay like O(exp (−cn1/(p + 1))) for a positive constant c.