Coanalytic models for Hardy-type operators

We establish coanalytic models for a broad class of Hardy-type operators on L2[0, 1]. In particular, we show that the logarithmic Hardy operator is unitarily equivalent to the difference between the identity operator and the backward shift on a Bergman-type space. This result leads to several applications related to zero sets and invariant subspaces in weighted Bergman spaces. Additionally, we study logarithmic Hardy operators on Lp[0, 1] and obtain results concerning their boundedness, operator norms, and spectra.