A Besov class functional calculus for bounded holomorphic semigroups

Abstract

It is well-known that π 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 [ ε , σ ] with 0 < ε < σ < ∞ . The constant bounding this calculus grows as log σ e ε as σ ε → ∞ and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that π 2 -sectorial operators admit a bounded calculus over the Besov algebra B ∞ 1 0 of the right half-plane. We also discuss the link between π 2 -sectorial operators and bounded Tadmor–Ritt operators.