Integration Operators on Bergman Spaces with exponential weight

Abstract

We study operators of the form $T_{g}f\left(z\right)={\int }_0^{z}f\left(\xi \right)g^{\prime }\left(\xi \right)d\left(\xi \right)$ ($g$ is an analytic function unity disc) on weighted Bergman spaces $L_a^p\left(w\right)$ of the unit disc where symbol $g$ is analytic function on the disc. For the case of

it is shown that operator $T_g$ is bounded (compact) on $L_a^2\left(w\right)$ if and only if ${\left(1-\left|z\right|\right)}^{\beta +1}\left|g^{\prime }\left(z\right)\right|=O\left(1\right)\left(=o\left(1\right)\right)$ as $\left|z\right|\to 1-$, thus solving a problem formulated in [Aleman, A. and Siskakis, A.G.: Integration Operators on Bergman Spaces. Indiana Univ. Math. J. 46 (1997), no. 2, 337-356.].