Contractivity of the H∞-calculus and Blaschke products

It is well known that a densely defined operator A on a Hilbert space is accretive if and only if A has a contractive H∞-calculus for any angle bigger than π 2 . A third equivalent condition is that ‖(A− w)(A + w)−1‖ ≤ 1 for all Rew ≥ 0. In the Banach space setting, accretivity does not imply the boundedness of the H∞-calculus any more. However, we show in this note that the last condition is still equivalent to the contractivity of the H∞calculus in all Banach spaces. Furthermore, we give a sufficient condition for the contractivity of the H∞-calculus on C+, thereby extending a Hilbert space result of Sz.-Nagy and Foiaş to the Banach space setting. Mathematics Subject Classification (2000). 47A60, 47B44, 30D50.