A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that π2-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 [ε,σ] with 0<ε<σ<∞. The constant bounding this calculus grows as logσeε as σε→∞ and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that π2-sectorial operators admit a bounded calculus over the Besov algebra B∞10 of the right half-plane. We also discuss the link between π2-sectorial operators and bounded Tadmor-Ritt operators.