  Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1<p<\infty. It
is known that if A and its adjoint A^* both satisfy square function estimates
\bignorm{\bigl(\int_{0}^{\infty}| A^{1/2} T_t(x)|^2\, dt\,\bigr)^{1/2}_{Lp}
\lesssim \norm{x} and \bignorm{\bigl(\int_{0}^{\infty}|A^{*}^{1/2}
T_t^*(y)|^2\, dt\,\bigr)^{1/2}_{Lp'} \lesssim \norm{y} for x in Lp and y in
Lp', then A admits a bounded H^{\infty}(\Sigma_\theta) functional calculus for
any \theta>\frac{\pi}{2}. We show that this actually holds true for some
\theta<\frac{\pi}{2}.
