一类加权卷积Fréchet代数
A class of weighted convolution Fréchet algebras
摘要：
对于$\mathbb R^+$上的代数权的增序列$(ω_n)$，我们研究了作为加权Banach代数$L^1(ω_n)$的交而得到的Fréchet代数$A(ω)=\bigcap_n L^1(ω_n)$的各种性质.我们证明了$A(ω)$的每个自同态都是标准的，如果对于所有n个\in \mathbb N $ there exists $ m \in \mathbb N $ such that $ ω_m（t）/ω_n（t）\to \infty $ as $ t \to \infty $. Moreover, we characterise the continuous derivations on this algebra: If for all $ n \in \mathbb N $ there exists $ m \in \mathbb N $ such that $ t*ω_n（t）/ω_m（t）$ is bounded on $ \mathbb R^+ $, then the continuous derivations on $ A（ω）$ are exactly the linear maps $ D $ of the form $ D（f）=（Xf）*μ $ for $ f \in A（ω）$, where $ μ $ is a measure in $ B（ω）= \bigcap_n M（ω_n）$ and $（Xf）（t）=tf（t）$ for $ t \in \mathbb R^+ $ and $ f \in A（ω）$. If the condition is not satisfied, we show that $ A（ω）$没有非零导数。
英文摘要：
For an increasing sequence $(ω_n)$ of algebra weights on $\mathbb R^+$ we study various properties of the Fréchet algebra $A(ω)=\bigcap_n L^1(ω_n)$ obtained as the intersection of the weighted Banach algebras $L^1(ω_n)$. We show that every endomorphism of $A(ω)$ is standard, if for all n\in\mathbb N$ there exists $m\in\mathbb N$ such that $ω_m(t)/ω_n(t)\to\infty$ as $t\to\infty$. Moreover, we characterise the continuous derivations on this algebra: If for all $n\in\mathbb N$ there exists $m\in\mathbb N$ such that $t*ω_n(t)/ω_m(t)$ is bounded on $\mathbb R^+$, then the continuous derivations on $A(ω)$ are exactly the linear maps $D$ of the form $D(f)=(Xf)*μ$ for $f\in A(ω)$, where $μ$ is a measure in $B(ω)=\bigcap_n M(ω_n)$ and $(Xf)(t)=tf(t)$ for $t\in\mathbb R^+$ and $f\in A(ω)$. If the condition is not satisfied, we show that $A(ω)$ has no non-zero derivations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
