Banach space operators with a boundedH∞ functional calculus

Abstract

AbstractIn this paper, we give a general definition forf(T)whenTis a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and whenfis holomorphic on a larger sector.We also examine how certain properties of this functional calculus, such as the existence of a boundedH∈functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, ifTis acting in a reflexiveLpspace, thenThas a boundedH∈ functional calculus if and only if bothTand 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.