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

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(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 mentioned above. We show that this is actually true if and only if Taylor coefficients f̂(n) of f decay like O(exp(−cn)) for a positive constant c.