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

TL;DR

This paper introduces a new algebra of analytic functions enabling a bounded functional calculus for generators of bounded $C_0$-semigroups on Hilbert spaces, improving previous calculus frameworks.

Contribution

It develops a novel Banach algebra ${ mf extit A}({ mf C}_+)$ and proves it provides a bounded functional calculus for generators of bounded $C_0$-semigroups on Hilbert spaces, surpassing earlier approaches.

Findings

Established a new algebra ${ mf extit A}({ mf C}_+)$ of bounded analytic functions.

Proved the generator of any bounded $C_0$-semigroup on Hilbert space admits a bounded functional calculus via ${ mf extit A}({ mf C}_+)$.

Compared and improved upon the existing calculus based on the algebra ${ mf B}({ mf C}_+)$.

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.