Let $\lbrace \beta (n)\rbrace ^{\infty }_{n=0}$ be a sequence of positive numbers and $1 \le p \infty $. We consider the space $H^{p}(\beta )$ of all power series $f(z)=\sum ^{\infty }_{n=0}\hat{f}(n)z^{n}$ such that $\sum ^{\infty }_{n=0}|\hat{f}(n)|^{p}\beta (n)^{p} \infty $. We investigate strict cyclicity of $H^{\infty }_{p}(\beta )$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^{p}(\beta )$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H^{\infty }_{p}(\beta )$. We also give a necessary condition for a composition operator to be bounded on $H^{p}(\beta )$ when $H^{\infty }_{p}(\beta )$ is strictly cyclic.
Let $\lbrace \beta (n)\rbrace ^{\infty }_{n=0}$ be a sequence of positive numbers and $1 \le p \infty $. We consider the space $H^{p}(\beta )$ of all power series $f(z)=\sum ^{\infty }_{n=0}\hat{f}(n)z^{n}$ such that $\sum ^{\infty }_{n=0}|\hat{f}(n)|^{p}\beta (n)^{p} \infty $. We investigate strict cyclicity of $H^{\infty }_{p}(\beta )$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^{p}(\beta )$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H^{\infty }_{p}(\beta )$. We also give a necessary condition for a composition operator to be bounded on $H^{p}(\beta )$ when $H^{\infty }_{p}(\beta )$ is strictly cyclic.
Classification :
46E15, 47A16, 47A25, 47B37
Keywords:
the Banach space of formal power series associated with a sequence $\beta $; bounded point evaluation; strictly cyclic maximal ideal space; Schatten $p$-class; reflexive algebra; semisimple algebra; composition operator
@article{CMJ_2004_54_1_a23,
author = {Yousefi, B.},
title = {Strictly cyclic algebra of operators acting on {Banach} spaces $H^p(\beta)$},
journal = {Czechoslovak Mathematical Journal},
pages = {261--266},
year = {2004},
volume = {54},
number = {1},
mrnumber = {2040238},
zbl = {1049.47033},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_1_a23/}
}
TY - JOUR
AU - Yousefi, B.
TI - Strictly cyclic algebra of operators acting on Banach spaces $H^p(\beta)$
JO - Czechoslovak Mathematical Journal
PY - 2004
SP - 261
EP - 266
VL - 54
IS - 1
UR - http://geodesic.mathdoc.fr/item/CMJ_2004_54_1_a23/
LA - en
ID - CMJ_2004_54_1_a23
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%0 Journal Article
%A Yousefi, B.
%T Strictly cyclic algebra of operators acting on Banach spaces $H^p(\beta)$
%J Czechoslovak Mathematical Journal
%D 2004
%P 261-266
%V 54
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2004_54_1_a23/
%G en
%F CMJ_2004_54_1_a23
Yousefi, B. Strictly cyclic algebra of operators acting on Banach spaces $H^p(\beta)$. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 261-266. http://geodesic.mathdoc.fr/item/CMJ_2004_54_1_a23/
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