It is well known that a densely defined operator $A$ on a Hilbert space is accretive if and only if A has a contractive $H^\infty$-calculus for any angle bigger than $\pi/2$. A third equivalent condition is that $(A − w)(A + w)^{−1} \leq 1$ for all $\Re w \geq 0$. 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 $H^\infty$ calculus in all Banach spaces. Furthermore, we give a sufficient condition for the contractivity of the $H^\infty$-calculus on $\C_+$, thereby extending a Hilbert space result of Sz.-Nagy and Foiaş to the Banach space setting.