On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions

Abstract

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators Ta in weighted sup-normed Banach spaces Hv∞ of holomorphic functions defined on the open unit disc D of the complex plane; both the weights v and symbols a are assumed to be radial functions on D. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol a such that Ta is not bounded in any space Hv∞ with an admissible weight v. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol a at the boundary of D guarantees the boundedness of Ta. The sufficient conditions for the boundedness and compactness of Ta, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition (B), which includes in addition to standard weight classes also many rapidly decreasing weights.