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 T a in weighted sup-normed Banach spaces H v ∞ 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 T a is not bounded in any space H v ∞ 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 T a . The sufficient conditions for the boundedness and compactness of T a , 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.