NORM OF THE HILBERT MATRIX OPERATOR ON THE WEIGHTED BERGMAN SPACES

We find the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space A(p,alpha) parallel to H parallel to(Ap,alpha -> Ap,alpha )>= pi/sin(alpha+2)pi/p, for 1 < alpha + 2 < p. We show that if 4 <= 2(alpha + 2) <= p, than parallel to H parallel to(Ap,alpha -> Ap,alpha) = pi/sin(alpha+2)pi/p, while if 2 <= alpha + 2 < p < 2(alpha + 2), upper bound for the norm parallel to H parallel to(Ap,alpha -> Ap,alpha), better then known, is obtained.