Compactness of resolvent operators generated by a class of composition semigroups on Hp
(1990) In Journal of Mathematical Analysis and Applications 147(1). p.171-179- 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 function-theoretic. In particular, the notion of a spiral-like function plays a key role. (Less)
Please use this url to cite or link to this publication:
https://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
- 0022-247X
- language
- English
- LU publication?
- no
- id
- f723ab61-30b4-439d-af64-716fef06a4de (old id 1467381)
- alternative location
- http://ida.lub.lu.se/cgi-bin/elsevier_local?YYUM0070-A-0022247X-V0147I01-9090391R
- date added to LUP
- 2016-04-01 15:28:40
- date last changed
- 2021-01-03 08:58:11
@article{f723ab61-30b4-439d-af64-716fef06a4de, 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.}}, author = {{Aleman, Alexandru}}, issn = {{0022-247X}}, language = {{eng}}, number = {{1}}, pages = {{171--179}}, publisher = {{Elsevier}}, series = {{Journal of Mathematical Analysis and Applications}}, title = {{Compactness of resolvent operators generated by a class of composition semigroups on Hp}}, url = {{http://ida.lub.lu.se/cgi-bin/elsevier_local?YYUM0070-A-0022247X-V0147I01-9090391R}}, volume = {{147}}, year = {{1990}}, }