On boundedness and compactness of Toeplitz operators in weighted H∞-spaces

Abstract

We characterize the boundedness and compactness of Toeplitz operators Ta with radial symbols a in weighted H∞-spaces Hv∞ on the open unit disc of the complex plane. The weights v are also assumed radial and to satisfy the condition (B) introduced by the second named author. The main technique uses Taylor coefficient multipliers, and the results are first proved for them. We formulate a related sufficient condition for the boundedness and compactness of Toeplitz operators in reflexive weighted Bergman spaces on the disc.We also construct a bounded harmonic symbol f such that Tf is not bounded in Hv∞ for any v satisfying mild assumptions. As a corollary, the Bergman projection is never bounded with respect to the corresponding weighted sup-norms. However, we also show that, for normal weights v, all Toeplitz operators with a trigonometric polynomial as the symbol are bounded on Hv∞.