On the generation of Arveson weakly continuous semigroups

Abstract

We consider here one-parameter semigroups $\mathbf{T}=(T(t){)}_{t>0}$ of bounded operators on a Banach space $X$ which are weakly continuous in the sense of Arveson. For such a semigroup $\mathbf{T}$ denote by ${\mathcal{M}}_{{\omega }_{\mathbf{T}}}$ the convolution algebra consisting in those measures $\mu$ on $(0,+\infty )$ such that ${\int }_0^{+\infty }\Vert T(t)\Vert d|\mu |(t)<+\infty .$ The Pettis integral ${\int }_0^{+\infty }T(t)d\mu (t)$ defines for $\mu \in {\mathcal{M}}_{{\omega }_{\mathbf{T}}}$ a bounded operator ${\varphi }_{\mathbf{T}}(\mu )$ on $X.$ Identifying the space $L_{{\omega }_{\mathbf{T}}}^1$ of (classes of) measurable functions $f$ satisfying ${\int }_0^{+\infty }|f(t)\Vert T(t)\Vert dt<+\infty$ to a closed subspace ${\mathcal{M}}_{{\omega }_{\mathbf{T}}}$ in the usual way, we define the Arveson ideal ${\mathcal{I}}_{\mathbf{T}}$ of the semigroup to be the closure in $\mathcal{B}(X)$ of ${\varphi }_{\mathbf{T}}(L_{{\omega }_{\mathbf{T}}}^1).$ Using a variant of a procedure introduced a long time ago by the author we introduce a dense ideal ${\mathcal{U}}_{\mathbf{T}}$ of ${\mathcal{I}}_{\mathbf{T}},$ which is a Banach algebra with respect to a suitable norm $\Vert .{\Vert }_{{\mathcal{U}}_{\mathbf{T}}},$ such that $\lim {\sup }_{t\to 0^{+}}\Vert T(t){\Vert }_{\mathcal{B}({\mathcal{U}}_{\mathbf{T}})}<+\infty .$ The normalized Arveson ideal ${\mathcal{J}}_{\mathbf{T}}$ is the closure of ${\mathcal{I}}_{\mathbf{T}}$ in $\mathcal{B}({\mathcal{U}}_{\mathbf{T}}).$ The Banach algebra ${\mathcal{J}}_{\mathbf{T}}$ has a sequential approximate identity and is isometrically isomorphic to a closed ideal of its multiplier algebra $\mathcal{M}({\mathcal{J}}_{\mathbf{T}}).$ The Banach algebras ${\mathcal{U}}_{\mathbf{T}},$ ${\mathcal{I}}_{\mathbf{T}}$ and ${\mathcal{J}}_{\mathbf{T}}$ are "similar", and the map $S_{u/v}\to S_{au/av}$ defines when $a$ generates a dense principal ideal of ${\mathcal{U}}_{\mathbf{T}}$ a pseudo bounded isomorphism from the algebre $\mathcal{Q}\mathcal{M}({\mathcal{J}}_{\mathbf{T}})$ of quasimultipliers on ${\mathcal{J}}_{\mathbf{T}}$ onto the quasimultipliers algebras $\mathcal{Q}\mathcal{M}({\mathcal{U}}_{\mathbf{T}})$ and $\mathcal{Q}\mathcal{M}({\mathcal{I}}_{\mathbf{T}}).$ We define the generator $A_{\mathbf{T}}$ of the semigroup $\mathbf{T}$ to be a quasimultiplier on ${\mathcal{I}}_{\mathbf{T}},$ or ,equivalently, on ${\mathcal{J}}_{\mathbf{T}}.$ Every character $\chi$ on ${\mathcal{I}}_{\mathbf{T}}$ has an extension $\tilde{\chi }$ to $\mathcal{Q}\mathcal{M}({\mathcal{I}}_{\mathbf{T}}).$ Let $Re{s}_{ar}(A_{\mathbf{T}})$ be the complement of the set $\{\tilde{\chi }(A_{\mathbf{T}}){\}}_{\chi \in \widehat{{\mathcal{I}}_{\mathbf{T}}}}.$ The quasimultiplier $A-\mu I$ has an inverse belonging to ${\mathcal{J}}_{\mathbf{T}}$ for $\mu \in Re{s}_{ar}(A_{\mathbf{T}}),$ which allows to consider this inverse as a "regular" quasimultiplier on the Arveson ideal ${\mathcal{I}}_{\mathbf{T}}.$ The usual resolvent formula holds in this context for $Re(\mu )>{\lim }_{t\to +\infty }\frac{log\Vert T(t)\Vert }{t}.$ Set ${\Pi }_{\alpha }^{+}:=\{z\in \mathbb{C}|Re(z)>\alpha \}.$ We revisit the functional calculus associated to the generator $A_{\mathbf{T}}$ by defining $F(-A_{\mathbf{T}})\in {\mathcal{J}}_{\mathbf{T}}$ by a Cauchy integral when $F$ belongs to the Hardy space $H^1({\Pi }_{\alpha }^{+})$ for some $\alpha <-{\lim }_{t\to +\infty }\frac{log\Vert T(t)|}{t}.$ We then define $F(-A_{\mathbf{T}})$ as a quasimultiplier on ${\mathcal{J}}_{\mathbf{T}}$ and ${\mathcal{I}}_{\mathbf{T}}$ when $F$ belongs to the Smirnov class on ${\Pi }_{\alpha }^{+},$ and $F(-A_{\mathbf{T}})$ is a regular quasimultiplier on ${\mathcal{J}}_{\mathbf{T}}$ and ${\mathcal{I}}_{\mathbf{T}}$ if $F$ is bounded on ${\Pi }_{\alpha }^{+}.$ If $F(z)=e^{-zt}$ for some $t>0,$ then $F(-A_{\mathbf{T}})=T(t),$ and if $F(z)=-z,$ we indeed have $F(-A_{\mathbf{T}})=A_{\mathbf{T}}.$