Compactness of resolvent operators generated by a class of composition semigroups on Hp
(1990) In Journal of Mathematical Analysis and Applications 147(1). p.171179 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... (More)
 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 functiontheoretic. In particular, the notion of a spirallike function plays a key role. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1467381
 author
 Aleman, Alexandru ^{LU}
 publishing date
 1990
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Journal of Mathematical Analysis and Applications
 volume
 147
 issue
 1
 pages
 171  179
 publisher
 Elsevier
 external identifiers

 scopus:0025399828
 ISSN
 0022247X
 language
 English
 LU publication?
 no
 id
 f723ab6130b4439daf64716fef06a4de (old id 1467381)
 alternative location
 http://ida.lub.lu.se/cgibin/elsevier_local?YYUM0070A0022247XV0147I019090391R
 date added to LUP
 20090916 14:10:08
 date last changed
 20180529 10:22:27
@article{f723ab6130b4439daf64716fef06a4de, 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 functiontheoretic. In particular, the notion of a spirallike function plays a key role.}, author = {Aleman, Alexandru}, issn = {0022247X}, language = {eng}, number = {1}, pages = {171179}, publisher = {Elsevier}, series = {Journal of Mathematical Analysis and Applications}, title = {Compactness of resolvent operators generated by a class of composition semigroups on Hp}, volume = {147}, year = {1990}, }