Contractivity of the H ^∞-calculus and Blaschke Products

Abstract

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 $\tfrac{\pi } {2}$. A third equivalent condition is that $\left\| {\left( {A - w} \right)\left( {A + \bar w} \right)^{ - 1} } \right\| \leqslant 1$ for all Re w≥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 Foias to the Banach space setting.