Functional calculus for a bounded $C_0$-semigroup on Hilbert space

Abstract: We introduce a new Banach algebra ${\mathcal A}({\mathbb C}+)$ of bounded analytic functions on ${\mathbb C}+={z\in{\mathbb C}\, :\, {\rm Re}(z)>0}$ which is an analytic version of the Figa-Talamenca-Herz algebras on ${\mathbb R}$. Then we prove that the negative generator $A$ of any bounded $C_0$-semigroup on Hilbert space $H$ admits a bounded (natural) functional calculus $\rho_A\colon {\mathcal A}({\mathbb C}+)\to B(H)$. We prove that this is an improvement of the bounded functional calculus ${\mathcal B}({\mathbb C}+)\to B(H)$ recently devised by Batty-Gomilko-Tomilov on a certain Besov algebra ${\mathcal B}({\mathbb C}+)$ of analytic functions on ${\mathbb C}+$, by showing that ${\mathcal B}({\mathbb C}+)\subset {\mathcal A}({\mathbb C}+)$ and ${\mathcal B}({\mathbb C}+)\not= {\mathcal A}({\mathbb C}+)$. In the Banach space setting, we give similar results for negative generators of $\gamma$-bounded $C_0$-semigroups. The study of ${\mathcal A}({\mathbb C}_+)$ requires to deal with Fourier multipliers on the Hardy space $H^1({\mathbb R})\subset L^1({\mathbb R})$ of analytic functions.