Banach space operators with a bounded H infinity functional calculus

In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when f is holomorphic on a larger sector. We also examine how certain properties of this functional calculus, such as the existence of a bounded H-infinity functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive L(p) space, then T has a bounded H-infinity functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.