Toeplitz operators and arguments of analytic functions

For a Toeplitz operator T _φ , we study the interrelationship between smoothness properties of the symbol φ and those of the functions annihilated by T _φ . For instance, it follows from our results that if φ is a unimodular function on the circle lying in some Lipschitz or Zygmund space Λ^α with 0 < α < ∞, and if f is an H ^p -function (p ≥ 1) with T _φ f = 0, then f ∈ Λ^α and for some c = c(α, p) and d = d(α, 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 ∈ H ^p lies in the kernel of with , 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. Supported in part by grants MTM2008-05561-C02-01/MTM, HF2006-0211 and MTM2007-30904-E from El Ministerio de Ciencia e Innovación (Spain), and by grant 2005-SGR-00611 from DURSI (Generalitat de Catalunya).