Arch. Math., Vol. 67, 312-318 (1996) 0003-889X/96/6704-03~2 $ 2.90/0 t996 Birkhfiuser VerIag, Basel On the Bergman space norm of the Cesfiro operator By ARISTOMENIS G. SISKAKIS I. Introduction. Let D denote the unit disc in the complex plane rE, and dm = (l/re) dx dy the normalized Lebesgue measure on ID. For I < p < ~ the Bergman space A p is the closed subspace of all analytic functions in Lp (ID, dm). For f analytic on ID the A p norm is (1.1) tlfll~, = j" If(z)Vdm(z), UD while for p = 2 we can use the expression (1.2) Ilf{[~ = Z ra,[2/(n + 1), n__>0 where f(z) = Z a,z". For such f analytic on ID the Ces~iro transformation C is defined by ,_>o (1.3) C (f) (z) = -z o - g ,= o ~ k~O Z". 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, 1i]. 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)-* (g) (z) = - z (1 - z) 9' (z) - (1 - z) 9 (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. Any semigroup of composition operators is also strongly continuous on the Bergman spaces A p, in fact on their weighted versions Ag with weights w (r) = (i - rZ)~. This was the main result of [12], along with the identification of the corresponding infinitesimal generators. In addition as an application we found in [12] that the operator (y~ ak ~zk where f(z)= Z ak zk, d(f)(z) Z ,,~=o \~=n k + l j k>=o is bounded on A~ if and only if c~ + 2 < p [12, Theorem 3]. In particular with • = 0 d is bounded on A p if and only ifp > 2. It is clear that the operators C and ~r are induced