POSITIVE OPERATORS ON THE BERGMAN SPACE AND BEREZIN TRANSFORM

Let D = {z is an element of C : vertical bar z vertical bar < 1} and L-a(2)(D) be the Bergman space of the disk. In this paper we characterize the class A subset of L-infinity (D) such that if phi, psi is an element of A, alpha >= 0 and 0 <= phi <= alpha psi then there exist positive operators S,T is an element of L(L-a(2) (D)) such that phi(z) = S(z) <= alpha T(z) = alpha psi(z) for all z is an element of D. Further, we have shown that if S and T are two positive operators in L (L-a(2)(D)) and T is invertible then there exists a constant a >= 0 such that (S) over tilde (z) <= a (T) over tilde (z) for all z is an element of D and (S) over tilde,(T) over tilde E A. Here (L-a(2)(D)) is the space of all bounded linear operators from L-a(2)(D)) into L-a(2)(D) and (A) over tilde (z) = (Akz, kz) is the Berezin transform of A is an element of L-a(2)(D)) and kz is the normalized reproducing kernel of L,i(1.D)). Applications of these results are also obtained.