Abstract view
Convergence Factors and Compactness in Weighted Convolution Algebras


Published:20020401
Printed: Apr 2002
Fereidoun Ghahramani
Sandy Grabiner
Abstract
We study convergence in weighted convolution algebras $L^1(\omega)$ on
$R^+$, with the weights chosen such that the corresponding weighted
space $M(\omega)$ of measures is also a Banach algebra and is the dual
space of a natural related space of continuous functions. We
determine convergence factor $\eta$ for which
weak$^\ast$convergence of $\{\lambda_n\}$ to $\lambda$ in $M(\omega)$
implies norm convergence of $\lambda_n \ast f$ to $\lambda \ast f$ in
$L^1 (\omega\eta)$. We find necessary and sufficent conditions which
depend on $\omega$ and $f$ and also find necessary and sufficent
conditions for $\eta$ to be a convergence factor for all $L^1(\omega)$
and all $f$ in $L^1(\omega)$. We also give some applications to the
structure of weighted convolution algebras. As a preliminary result
we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if
and only if convolution by $f$ is a compact operator from $M(\omega)$
(or $L^1(\omega)$) to $L^1 (\omega\eta)$.
MSC Classifications: 
43A10, 43A15, 46J45, 46J99 show english descriptions
Measure algebras on groups, semigroups, etc. $L^p$spaces and other function spaces on groups, semigroups, etc. Radical Banach algebras None of the above, but in this section
43A10  Measure algebras on groups, semigroups, etc. 43A15  $L^p$spaces and other function spaces on groups, semigroups, etc. 46J45  Radical Banach algebras 46J99  None of the above, but in this section
