论文标题
Dirichlet系列耐力空间的组成操作员的半群
Semigroups of composition operators on Hardy spaces of Dirichlet series
论文作者
论文摘要
We consider continuous semigroups of analytic functions $\{Φ_t\}_{t\geq0}$ in the so-called Gordon-Hedenmalm class $\mathcal{G}$, that is, the family of analytic functions $Φ:\mathbb C_+\to \mathbb C_+$ giving rise to bounded composition operators in the Hardy space dirichlet系列$ \ mathcal {h}^2 $。我们表明，在类$ \ Mathcal g $中，连续的半群之间存在一对一的对应关系。 $ f \ in \ Mathcal {H}^2 $。我们将这些结果扩展到[1，\ infty）$的范围$ p \。对于$ p = \ infty $的情况，我们证明，在$ \ Mathcal {H}^\ infty $中没有非平凡的构图运算符。我们将类$ \ Mathcal g $连续半群的无限发电机表征为那些dirichlet系列，将$ \ mathbb c _ {+} $发送到其关闭中。半群的某些动力学特性是从Semigroup的Koenigs图的描述中获得的。
We consider continuous semigroups of analytic functions $\{Φ_t\}_{t\geq0}$ in the so-called Gordon-Hedenmalm class $\mathcal{G}$, that is, the family of analytic functions $Φ:\mathbb C_+\to \mathbb C_+$ giving rise to bounded composition operators in the Hardy space of Dirichlet series $\mathcal{H}^2$. We show that there is a one-to-one correspondence between continuous semigroups $\{Φ_{t}\}_{t\geq0}$ in the class $\mathcal G$ and strongly continuous semigroups of composition operators $\{T_t\}_{t\geq0}$, where $T_t(f)=f\circΦ_t$, $f\in\mathcal{H}^2$. We extend these results for the range $p\in[1,\infty)$. For the case $p=\infty$, we prove that there is no non-trivial strongly continuous semigroup of composition operators in $\mathcal{H}^\infty$. We characterize the infinitesimal generators of continuous semigroups in the class $\mathcal G$ as those Dirichlet series sending $\mathbb C_{+}$ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.
