Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494220230301Bergman and Dirichlet spaces in the unit ball and symmetric lifting operator155160500510.22060/ajmc.2022.21778.1107ENMostafaHassanlouEngineering Faculty of Khoy, Urmia University of Technology, Urmia, IranEbrahimAbbasiDepartment of Mathematics, Mahabad Branch, Islamic Azad University, Mahabad, IranJournal Article20220916Let $\mathbb{B}_n$ be the open unit ball in $\mathbb{C}^n$ and $\mathbb{B}_n^2 = \mathbb{B}_n \times \mathbb{B}_n$. The symmetric lifting operator which lifts analytic functions from $H(\mathbb{B}_n)$ to $H(\mathbb{B}_n^2)$ is defined as follow<br />\[<br />L(f)(z,w) = \frac{f(z) - f(w)}{z-w}.<br />\]<br />In this paper we investigate the action of symmetric lifting operator on the Bergman space in the unit ball. Also, we state a characterization for Dirichlet space and consider symmetric lifting operator on the Dirichlet space in the unit ball.https://ajmc.aut.ac.ir/article_5005_406b2153e9a979a51060d31b7bcbf7cc.pdf