Bergman and Dirichlet spaces in the unit ball and symmetric lifting operator

Let Bn be the open unit ball in Cn and Bn×Bn. The symmetric lifting operator which lifts analytic functions from H(Bn) to H(Bn2) is defined as follow‎‎                                                                         L(f)(z,w) = f(z)-f(w)/z-w.‎‎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.