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 $H_v^{\infty }$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. 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 $H_v^{\infty }$ 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 $\mathbb{D}$ 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.