The dual of a generalized weighted Bergman space

The generalized weighted Bergman space H(B-d, lambda) is defined as a reproducing kernel Hilbert space of holomorphic functions on the open unit ball B-d subset of C-d for all lambda > 0. When lambda > d, it is identical to the weighted Bergman space HL2(B-d, mu(lambda)). We prove that the dual space H(B-d, alpha)* can be identified with another generalized weighted Bergman space H(B-d, beta) under the pairing < f, g >(gamma) = integral(Bd) A(lambda)f(x)<(B(lambda)g(z))over bar>d mu(gamma+2n)(z), for f is an element of H(B-d, alpha), g is an element of H(B-d, beta), where n = [d/2], gamma = alpha+beta/2 and A(lambda), B-lambda are operators related to the number operator N =Sigma(d)(i=1) z(i) partial derivative/partial derivative(zi).