STUDIA MATHEMATICA 196 (3) (2010) The one-sided ergodic Hilbert transform in Banach spaces by Guy Cohen (Beer Sheva), Christophe Cuny (Noum´ ea) and Michael Lin (Beer Sheva) Abstract. Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform limn P n k=1 T k x k . We prove that weak and strong convergence are equivalent, and in a reflexive space also sup n ‖ P n k=1 T k x k ‖ < ∞ is equivalent to the convergence. We also show that − P ∞ k=1 T k k (which converges on (I − T )X) is precisely the infinitesimal generator of the semigroup (I − T ) r | (I-T )X . 1. Introduction. Izumi [12] 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 transformation θ and functions in L 2 (S,Σ,µ) (which, by Kronecker’s lemma, would be a strengthening of Birkhoff’s pointwise er- godic theorem). Halmos [10] proved that for every ergodic probability pre- serving transformation on a non-atomic space there always exists a centred f ∈ L 2 such that the one-sided EHT fails to converge in L 2 -norm. On the other hand, Cotlar [5] proved that when T is the operator induced by an invertible probability preserving transformation, for every f ∈ L 1 the two-sided EHT ∑ ∞ k=1 T k f −T -k f k converges a.e., and in L p -norm when f ∈ L p , 1 ≤ p< ∞. Campbell [2] proved that for a unitary operator T on a complex Hilbert space H the two-sided EHT converges in norm for every f ∈ H . For T unitary on a complex Hilbert space H , Gaposhkin [9] obtained a spectral characterization of the norm convergence of the one-sided EHT ∑ ∞ k=1 T k f k . For a normal contraction T , several additional characterizations were recently obtained by Cohen and Lin [3], who proved that norm con- vergence is equivalent to weak convergence; this had been proved by Assani and Lin [1] for T unitary or self-adjoint. 2010 Mathematics Subject Classification : Primary 47A35, 28D05, 37A05; Secondary 47B38. Key words and phrases : ergodic Hilbert transform, operator power series, measure pre- serving transformations, semigroup of fractional powers. DOI: 10.4064/sm196-3-3 [251] c  Instytut Matematyczny PAN, 2010