# Integration Operators on Bergman Spaces with exponential weight

### Milutin R. Dostanić

University of Belgrade, Beograd, Serbia

## Abstract

We study operators of the form $T_{g}f(z)=∫_{0}f(ξ)g_{′}(ξ)d(ξ)$ ($g$ is an analytic function unity disc) on weighted Bergman spaces $L_{a}(w)$ of the unit disc where symbol $g$ is analytic function on the disc. For the case of

$w(r)=exp((1−r)_{β}−a )(a>0,0<β≤1)$

it is shown that operator $T_{g}$ is bounded (compact) on $L_{a}(w)$ if and only if $(1−∣z∣)_{β+1}∣g_{′}(z)∣=O(1)(=o(1))$ as $∣z∣→1−$, 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.].

## Cite this article

Milutin R. Dostanić, Integration Operators on Bergman Spaces with exponential weight. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 421–436

DOI 10.4171/RMI/501