Toeplitz operators and arguments of analytic functions

For a Toeplitz operator T-phi we study the interrelationship between smoothness properties of the symbol phi and those of the functions annihilated by T-phi. For instance, it follows from our results that if phi is a unimodular function on the circle lying in some Lipschitz or Zygmund space A(alpha) with 0 < alpha < infinity, and if f is an H-p-function (p >= 1) with T-phi f = 0, then f is an element of Lambda(alpha) and parallel to f parallel to Lambda(alpha) <= c parallel to phi parallel to(d)(Lambda alpha)parallel to f parallel to p for some c = c(alpha, p) and d = d(alpha, p); an explicit formula for the optimal exponent d is provided. Similar - and more general - results for various smoothness classes are obtained, and several approaches are discussed. Furthermore, since a given non-null function f is an element of H-p lies in the kernel of T-psi with psi = (z) over bar (f) over bar /f, we derive information on the smoothness of H-p-functions with smooth arguments. This can be viewed as a natural counterpart to the existing theory of analytic functions with smooth moduli.