On the generation of Arveson weakly continuous semigroups

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 $F(-A_{\bf T})=A_{\bf T}.$