bergman and dirichlet spaces in the unit ball and symmetric lifting operator

Let bn be the open unit ball in c n and b 2 n = bn×bn. the symmetric lifting operator which lifts analytic functions from h(bn) to h(b 2 n ) 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.