The $H^{\mathrm{\infty }}$-Functional Calculus and Square Function Estimates

Using notions from the geometry of Banach spaces we introduce square functions $\gamma (\mathrm{\Omega },X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(\mathrm{\Omega },H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $\gamma (\mathrm{\Omega },X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\mathrm{\infty }}$–calculus that extend known results for $L_p$–spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e. g., that a $c_0$–group of operators $T_s$ on a Banach space with finite cotype has an $H^{\mathrm{\infty }}$–calculus on a strip if and only if $e^{-a\mathrm{\mid }s\mathrm{\mid }}{T}_s$ is $R$–bounded for some $a>0$. Similarly, a sectorial operator $A$ has an $H^{\mathrm{\infty }}$–calculus on a sector if and only if $A$ has $R$–bounded imaginary powers. We also consider vector valued Paley–Littlewood $g$–functions on $UMD$–spaces.