HILBERT MATRIX OPERATOR ON SPACES OF ANALYTIC FUNCTIONS

We consider the action of the Hilbert matrix operator, H, On the Hardy space H-1, weighted Hardy spaces H-alpha(p) (alpha >= 0), Bergman spaces with logarithmic weights, etc. In particular, we extend Diamantopoulos-Siskakis result by proving that H maps H-alpha(p) into H-alpha(p) if and only if alpha+1/p < 1. A criterion for Hf to belong to H-1 is given provided the coefficients of f are nonnegative. Also, H maps the A(2)-space with weight log(alpha)(2/(1 - vertical bar z vertical bar(2))) into the ordinary Bergman space A(2) if alpha > 3. Similarly, the Bloch space with logarithmic weight is mapped by H into the ordinary Bloch space.