Let T: Lp (Ω) → Lp (Ω) be a contraction, with 1< p < ∞, and assume that T is analytic, that is, supn⩾1 n‖Tn – Tn–1‖ < ∞. 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 ∥ ( Σ n = 1 ∞ n 2 m − 1 | T n ( T − 1 ) m ( x ) | 2 ) 1 / 2 ∥ p ≲ ∥ x ∥ p for any integer m ⩾1. As a consequence, we show maximal inequalities of the form ∥ sup n ⩾ 0 ( n + 1 ) m | T n ( T − I ) m ( x ) | ∥ p ≲ ∥ x ∥ p , for any integer m ⩾ 0. We prove similar results in the context of noncommutative Lp-spaces. We also give analogs of these maximal inequalities for bounded analytic semigroups, as well as applications to R-boundedness properties.