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

TL;DR

This paper introduces new square function tools for Banach space-valued functions and uses them to characterize the $H^{ abla}$-calculus for operators, extending known results from $L_p$ spaces.

Contribution

It develops a general framework for square functions in Banach spaces and applies it to extend $H^{ abla}$-calculus characterizations beyond classical $L_p$ settings.

Findings

Characterization of $H^{ abla}$-calculus via $R$-boundedness.

Extension of square functions to arbitrary Banach spaces.

Application to $c_0$-groups and sectorial operators.

Abstract

Using notions from the geometry of Banach spaces we introduce square functions $\gamma(\Omega,X)$ 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 $L_2(\Omega,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $\gamma(\Omega,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--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 $H^{\infty}$--calculus on a strip if and only if $e^{-a|s|}T_s$ is $R$--bounded for some $a > 0$. Similarly, a sectorial operator $A$ has an $H^{\infty}$--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.