Functional calculus for a bounded $C_0$-semigroup on Hilbert space
Loris Arnold, Christian Le Merdy
TL;DR
The paper constructs a new analytic Banach algebra $\mathcal{A}(\mathbb{C}_+)$ that serves as an analytic analogue of Figa-Talamanca-Herz algebras and proves that the negative generator $A$ of a bounded $C_0$-semigroup on a Hilbert space admits a bounded functional calculus $\rho_A: \mathcal{A}(\mathbb{C}_+) \to B(H)$. This calculus is shown to improve the Besov calculus of Batty-Gomilko-Tomilov by embedding $\mathcal{B}_0(\mathbb{C}_+)$ into $\mathcal{A}(\mathbb{C}_+)$ with a quantitative bound, and it also extends to $\gamma$-bounded semigroups on Banach spaces, where the framework relies on Fourier multipliers on $H^1(\mathbb{R})$. The authors develop half-plane holomorphic calculus, establish duality and tensor-product structures for $\mathcal{A}_0$ and $\mathcal{A}$, and relate their calculus to $H^\infty(\mathbb{C}_+)$-calculus. A May 2022 addendum strengthens the duality with $\mathcal{M}(H^1(\mathbb{R}))$ and discusses refinements via $S^1$-bounded multipliers.
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 $ρ_A\colon {\mathcal A}({\mathbb C}_+)\to B(H)$. 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 $γ$-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.