Composition operators on Banach spaces of formal power series

Yousefi, B., and Jahedi, S.. "Composition operators on Banach spaces of formal power series." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 481-487. <http://eudml.org/doc/195221>.

@article{Yousefi2003,
abstract = {Let $\\{\beta (n)\\}^\{\infty\}_\{n=0\}$ be a sequence of positive numbers and $1\leq p < \infty$. We consider the space $H^\{p\}(\beta)$ of all power series $f(z)= \sum_\{n=0\}^\{\infty\} \hat\{f\}(n)z^\{n\}$ such that $\sum_\{n=0\}^\{\infty\}|\hat\{f\}(n)|^\{p\}\beta(n)^\{p\}<\infty $ . Suppose that $\frac\{1\}\{p\}+\frac\{1\}\{q\}=1$ and $\sum_\{n=1\}^\{\infty\} \frac\{n^\{qj\}\}\{\beta(n)^\{q\}\}=\infty$ for some nonnegative integer $j$. We show that if $C_\{\varphi\}$ is compact on $H^\{p\}(\beta)$, then the non-tangential limit of $\varphi^\{(j+1)\}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_\{\varphi\}$ is Fredholm on $H_\{p\}(\beta)$, then $\varphi$ must be an automorphism of the open unit disc.},
author = {Yousefi, B., Jahedi, S.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {481-487},
publisher = {Unione Matematica Italiana},
title = {Composition operators on Banach spaces of formal power series},
url = {http://eudml.org/doc/195221},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Yousefi, B.
AU - Jahedi, S.
TI - Composition operators on Banach spaces of formal power series
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 481
EP - 487
AB - Let $\{\beta (n)\}^{\infty}_{n=0}$ be a sequence of positive numbers and $1\leq p < \infty$. We consider the space $H^{p}(\beta)$ of all power series $f(z)= \sum_{n=0}^{\infty} \hat{f}(n)z^{n}$ such that $\sum_{n=0}^{\infty}|\hat{f}(n)|^{p}\beta(n)^{p}<\infty $ . Suppose that $\frac{1}{p}+\frac{1}{q}=1$ and $\sum_{n=1}^{\infty} \frac{n^{qj}}{\beta(n)^{q}}=\infty$ for some nonnegative integer $j$. We show that if $C_{\varphi}$ is compact on $H^{p}(\beta)$, then the non-tangential limit of $\varphi^{(j+1)}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_{\varphi}$ is Fredholm on $H_{p}(\beta)$, then $\varphi$ must be an automorphism of the open unit disc.
LA - eng
UR - http://eudml.org/doc/195221
ER -