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

[EN] We prove sufficient conditions for the boundedness and compactness of Toeplitz operators T-a in weighted sup-normed Banach spaces H-v(infinity) 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 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(infinity) 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. (c) 2020 Elsevier Inc. All rights reserved.