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

Aleman, Alexandru LU (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)
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author
publishing date
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
2009-09-16 14:10:08
date last changed
2017-01-01 06:40:27
@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 $H^p$ $(1\leq p&lt;\infty)$ 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 $q&lt;\infty$. 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},
  volume       = {147},
  year         = {1990},
}