On the Bergman space norm of the Cesaro operator

摘要

1. Introduction. Let D denote the unit disc in the complex plane (C, and dm = (l)dxdy the normalized Lebesgue measure on D. For 1 g p < co the Bergman space A~n is the closed subspace of all analytic functions in If (ID, dm). For f analytic on ID the Ap norm is. while for p = 2 we can use the expression The averaging operator C and its continuous analogues have been studied on various spaces including sequence spaces and the Hardy spaces [1, 2, 3, 7, 9, 11], In the case of Hardy spaces, C has been related to a semigroup of composition operators [2, 3, 11], thereby giving a method of studying C by studying the semigroup. The observation providing this link is that on the space of all analytic functions on D, (-C)~l (g) (z) = - z(l - z) g'(z) - (1 - z)g(z), and the restriction of this differential operator on Hardy spaces is found to be the infinitesimal generator of a specific strongly continuous compo-sition semigroup.