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 linea r operator acting in a Banach space, whose spectrum lies within some s ector, and which satisfies certain resolvent bounds, and when f is hol omorphic on a larger sector. We also examine how certain properties of this functional calculus, such as the existence of a bounded H-infini ty functional calculus, bounds on the imaginary powers, and square fun ction estimates are related. In particular we show that, if T is actin g in a reflexive L(p) space, then T has a bounded H-infinity functiona l 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 h old for operators in a Hilbert space do not extend to the general Bana ch space setting.