Convergence Factors and Compactness in Weighted Convolution Algebras

Ghahramani, Fereidoun ; Grabiner, Sandy

Canadian journal of mathematics, Tome 54 (2002) no. 2, pp. 303-323

Résumé

We study convergence in weighted convolution algebras ${{L}^{1}}\left( \omega\right)$ on ${{R}^{+}}$ , with the weights chosen such that the corresponding weighted space $M\left( \omega\right)$ 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 $*$ -convergence of $\left\{ {{\text{ }\!\!\lambda\!\!\text{ }}_{n}} \right\}$ to $\text{ }\!\!\lambda\!\!\text{ }$ in $M\left( \omega\right)$ implies norm convergence of ${{\text{ }\!\!\lambda\!\!\text{ }}_{n}}*f$ to $\text{ }\!\!\lambda\!\!\text{ *}f$ in ${{L}^{1}}\left( \omega \eta\right)$ . 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 ${{L}^{1}}\left( \omega\right)$ and all $f$ in ${{L}^{1}}\left( \omega\right)$ . 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 $M\left( \omega\right)$ (or ${{L}^{1}}\left( \omega\right)$ ) to ${{L}^{1}}\left( \omega \eta\right)$ .

Détail

DOI : 10.4153/CJM-2002-010-5

Mots-clés : 43A10, 43A15, 46J45, 46J99

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