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 ${\mathbb{C}}_{+}=\{z\in \mathbb{C}:\mathrm{Re}(z)>0\}$ 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 $\ddot{I}{}_A:\mathcal{A}({\mathbb{C}}_{+})\to B(H)$. 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 $\mathcal{B}({\mathbb{C}}_{+})\ne \mathcal{A}({\mathbb{C}}_{+})$. In the Banach space setting, we give similar results for negative generators of ${\hat{I}}^3$-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.

This updated version is published in Journal of Functional Analysis