Compactness of resolvent operators generated by a class of composition semigroups on Hp

Abstract

Consider a semigroup of composition operators $T_tf=f\circ \phi_t$, $t\geq 0$, acting on the standard Hardy space $H^p$ $(1\leq p<\infty)$ of the unit disk $D$. Assuming that the $\phi_t$ have a common fixed point at 0, a unique univalent function $h$ can be found such that $h(\phi_t(z))=e^{ct}h(z)$, $z\in D$, $t\geq 0$, where $c$ is a constant which is related to the infinitesimal generator $A$ of $\{T_t\}$. In this paper the compactness of the resolvent operator $R(\lambda,A)$ is studied using the function $h$. It is shown that $R(\lambda,A)$ is compact if and only if $h$ lies in $H^q$ for each $q<\infty$. In the cases where $R(\lambda,A)$ is not compact it is shown that the spectrum of $R(\lambda,A)$ contains a nontrivial disk. A condition under which compactness of $R(\lambda,A)$ implies compactness of the operators in $\{T_t\}$ is also obtained. The methods used are function-theoretic. In particular, the notion of a spiral-like function plays a key role. (Less)