ON CONVERGENCE OF POWER SERIES OF L p CONTRACTIONS GUY COHEN, CHRISTOPHE CUNY, AND MICHAEL LIN Abstract. Let T be a power-bounded operator on a (real or com- plex) Banach space. We study the convergence of the power series ∑ ∞ k=0 β k T k x when {β k } is a Kaluza sequence with divergent sum such that β k → 0 and ∑ ∞ k=0 β k z k converges in the open unit disk. We prove that weak and strong convergence are equivalent, and in a reflexive space also sup n ‖ ∑ n k=0 β k T k x‖ < ∞ is equivalent to the convergence of the series. The last assertion is proved also when T is a mean ergodic con- traction of L 1 . For normal operators on a Hilbert space we obtain a spectral charac- terization of the convergence of ∑ ∞ n=0 β n T n x, and a sufficient condition expressed in terms of norms of the ergodic averages, which in some cases is also necessary. For T Dunford-Schwartz of a σ-finite measure space or a positive con- traction of L p ,1 <p< ∞, we prove that when {β k } is also completely monotone (i.e. a Hausdorff moment sequence) and β k = O(1/k), the norm convergence of ∑ ∞ k=0 β k T k f implies a.e. convergence. For T a positive contraction of L p , p> 1 and f ∈ L p , we show that if the series ∑ ∞ n=0 (log(n+1)) β (n+1) 1-1/r T n f converges in L p -norm for some r ∈ ( p p-1 , ∞], β ∈ R, then it converges a.e. 1. Introduction Izumi [31] raised the question of the a.e. convergence of the one-sided ergodic Hilbert transform (EHT) ∑ ∞ k=1 f ◦θ k k associated to a probability preserving ergodic transformation θ and centered functions in L 2 (S, Σ,µ) (which, by Kronecker’s lemma, would be a strengthening of Birkhoff’s point- wise ergodic theorem). Halmos [28] proved that for every ergodic probability preserving transformation on a non-atomic space there always exists a cen- tered f ∈ L 2 such that the one-sided EHT fails to converge in L 2 -norm. Dowker and Erd¨os [20] (see also Del Junco and Rosenblatt [32]) obtained even the existence of f ∈ L ∞ (X ), centered, such that sup n≥1 | ∑ n k=1 f ◦θ k k | = +∞ a.s.; see [4] for additional background and references. 1991 Mathematics Subject Classification. Primary: 47A35, 28D05, 37A05; Secondary: 47B38. Key words and phrases. Ergodic Hilbert transform, operator power series, power- bounded operators, normal contractions, measure preserving tranformations, Dunford- Schwartz operators, positive contractions in L p . 1