INTEGRATION OPERATORS ON BERGMAN SPACES

We study operators of the form T-g(f)(z) =integral(0)(z)f(zeta)g'(zeta )d zeta, on weighted Bergman spaces L-a(p)(omega) of the unit disc. Fo r a large class of weights we show that T-g, is bounded on Lp(a)(p)(om ega) if and only if g is in the Bloch space, and compact if and only i f g is in the little Bloch space. For the weights omega(r) = (1-r)(alp ha), alpha > -1, we show that T-g is in the Schatten p-class of L-a(2) (omega) with p > 1 if and only if g is in the analytic p-Besov space.