Maximal theorems and square functions for analytic operators on Lp-spaces

Let T: L-p (Omega) -> L-p (Omega) be a contraction, with 1 < p < infinity, and assume that T is analytic, that is, sup(n >= 1) n vertical bar T-n - Tn-1 vertical bar < infinity. Under the assumption that T is positive (or contractively regular), we establish the boundedness of various Littlewood-Paley square functions associated with T. In particular, we show that T satisfies an estimate parallel to(Sigma(infinity)(n=1)n(2m-1)vertical bar T-m(T - I)(m)(x)vertical bar(2))(1/2)parallel to(p) less than or similar to parallel to x parallel to(p) for any integer m >= 1. As a consequence, we show maximal inequalities of the form parallel to sup(n >= 0)(n + 1)(m)vertical bar Tn(T - I)(m)(x)vertical bar parallel to(p) less than or similar to parallel to x parallel to(p), for any integer m >= 0. We prove similar results in the context of noncommutative L-p-spaces. We also give analogs of these maximal inequalities for bounded analytic semigroups, as well as applications to R-boundedness properties.