Functional calculus for a bounded C0-semigroup on Hilbert space

Abstract

We introduce a new Banach algebra A(C+) of bounded analytic functions on C+={z∈C:Re(z)>0} which is an analytic version of the Figa-Talamanca-Herz algebras on R. Then we prove that the negative generator A of any bounded C0-semigroup on Hilbert space H admits a bounded (natural) functional calculus ρA:A(C+)→B(H). We prove that this is an improvement of the bounded functional calculus B0(C+)→B(H) recently devised by Batty-Gomilko-Tomilov on a certain Besov algebra B0(C+) of analytic functions on C+, by showing that B0(C+)⊂A(C+) and B0(C+)≠A(C+). In the Banach space setting, we give similar results for negative generators of γ-bounded C0-semigroups. The study of A(C+) involves dealing with Fourier multipliers on the Hardy space H1(R)⊂L1(R) of analytic functions.