A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that 2 -sectorial operators generally do not admit a bounded H∞ calculus over the right half-plane. In contrast to this, we prove that the H∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval [ , ] with 0< < <∞. The constant bounding this calculus grows as log e as →∞ and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that 2 -sectorial operators admit a bounded calculus over the Besov algebra B0∞ 1 of the right half-plane. We also discuss the link between 2-sectorial operators and bounded Tadmor–Ritt operators. © 2005 Elsevier Inc. All rights reserved.