On the generation of Arveson weakly continuous semigroups

TL;DR

This paper studies weakly continuous semigroups of bounded operators on Banach spaces, introducing associated Banach algebras and ideals to analyze their structure and properties.

Contribution

It develops a framework of Banach algebras and ideals related to Arveson weakly continuous semigroups, including the construction of a dense ideal and the analysis of their multiplier algebras.

Findings

Introduces the Arveson ideal and normalized Arveson ideal for such semigroups.

Establishes isometric isomorphisms between related Banach algebras and their multiplier algebras.

Shows the existence of a sequential approximate identity in the normalized Arveson ideal.

Abstract

We consider here one-parameter semigroups ${\bf T}=(T(t))_{t>0}$ of bounded operators on a Banach space $X$ which are weakly continuous in the sense of Arveson. For such a semigroup ${\bf T}$ denote by ${\mathcal M}_{\omega_{\bf T}}$ the convolution algebra consisting in those measures $\mu$ on $(0,+\infty)$ such that $\int_0^{+\infty}\Vert T(t)\Vert d\vert \mu \vert (t)<+\infty.$ The Pettis integral $\int_0^{+\infty}T(t)d\mu(t)$ defines for $\mu \in {\mathcal M}_{\omega_{\bf T}}$ a bounded operator $\phi_{\bf T}(\mu)$ on $X.$ Identifying the space $L^1_{\omega_{\bf T}}$ of (classes of) measurable functions $f$ satisfying $\int_0^{+\infty}\vert f(t)\Vert T(t)\Vert dt< +\infty$ to a closed subspace ${\mathcal M}_{\omega_{\bf T}}$ in the usual way, we define the Arveson ideal $\mathcal{I}_{\bf T}$ of the semigroup to be the closure in ${\mathcal B}(X)$ of $\phi_{\bf T}(L^1_{\omega_{\bf T}}).$ Using a variant of a procedure introduced a long time ago by the author we introduce a dense ideal $\mathcal{U}_{\bf T}$ of $\mathcal{I}_{\bf T},$ which is a Banach algebra with respect to a suitable norm $\Vert .\Vert_{\mathcal{U}_{\bf T}},$ such that $\lim \sup_{t\to 0^+}\Vert T(t)\Vert_{{\mathcal B}(\mathcal{U}_{\bf T})}<+\infty.$ The normalized Arveson ideal $\mathcal{J}_{\bf T}$ is the closure of $\mathcal{I}_{\bf T}$ in ${\mathcal B}(\mathcal{U}_{\bf T}).$ The Banach algebra $\mathcal{J}_{\bf T}$ has a sequential approximate identity and is isometrically isomorphic to a closed ideal of its multiplier algebra ${\mathcal M}(\mathcal{J}_{\bf T}).$ The Banach algebras $\mathcal{U}_{\bf T},$ $\mathcal{I}_{\bf T}$ and $\mathcal{J}_{\bf T}$ are "similar", and the map $S_{u/v}\to S_{au/av}$ defines when $a$ generates a dense principal ideal of $\mathcal{U}_{\bf T}$ a pseudo bounded isomorphism from the algebre $\mathcal{QM}(\mathcal{J}_{\bf T})$ of quasimultipliers on $\mathcal{J}_{\bf T}$ onto the quasimultipliers algebras $\mathcal{QM}(\mathcal{U}_{\bf T})$ and $\mathcal{QM}(\mathcal{I}_{\bf T}).$ We define the generator $A_{\bf T}$ of the semigroup $\bf T$ to be a quasimultiplier on $\mathcal{I}_{\bf T},$ or ,equivalently, on $\mathcal{J}_{\bf T}.$ Every character $\chi$ on $\mathcal{I}_{\bf T}$ has an extension $\tilde \chi$ to $\mathcal{QM}(\mathcal{I}_{\bf T}).$ Let $Res_{ar} (A_{\bf T})$ be the complement of the set $\{ \tilde \chi (A_{\bf T})\}_{\chi \in \widehat{\mathcal{I}_{\bf T}}}.$ The quasimultiplier $A-\mu I$ has an inverse belonging to $\mathcal{J}_{\bf T}$ for $\mu \in Res_{ar} (A_{\bf T}),$ which allows to consider this inverse as a "regular" quasimultiplier on the Arveson ideal $\mathcal{I}_{\bf T}.$ The usual resolvent formula holds in this context for $Re(\mu)>\lim_{t\to +\infty}{log \Vert T(t)\Vert\over t}.$ Set $\Pi_{\alpha}^+:=\{ z \in \mathbb{C} \ | \ Re(z) >\alpha\}.$ We revisit the functional calculus associated to the generator $A_{\bf T}$ by defining $F(-A_{\bf T})\in \mathcal{J}_{\bf T}$ by a Cauchy integral when $F$ belongs to the Hardy space $H^1(\Pi_{\alpha}^+)$ for some $\alpha < -\lim_{t\to +\infty} {log\Vert T(t)\vert\over t}.$ We then define $F(-A_{\bf T})$ as a quasimultiplier on $\mathcal{J}_{\bf T}$ and $\mathcal{I}_{\bf T}$ when $F$ belongs to the Smirnov class on $\Pi_{\alpha}^+,$ and $F(-A_{\bf T})$ is a regular quasimultiplier on $\mathcal{J}_{\bf T}$ and $\mathcal{I}_{\bf T}$ if $F$ is bounded on $\Pi_{\alpha}^+.$ If $F(z)=e^{-zt}$ for some $t>0,$ then $F(-A_{\bf T})=T(t),$ and if $F(z)=-z,$ we indeed have $F(-A_{\bf T})=A_{\bf T}.$