Analytic Models for Hardy Type Operators

We establish the analytic models for a large class of Hardy type operators on L2[0,1]. As a specific example, it is shown that the logarithmic Hardy operator is unitarily equivalent to I-Mz* on the weighted Bergman space. Some applications concerning the zero sets and the invariant subspaces for the weighted Bergman spaces are discussed. Moreover, we study the logarithmic Hardy operators on Lp[0,1] and obtain some results on boundedness, operator norm and spectrum.