A class of weighted convolution Fr\'echet algebras

For an increasing sequence $(\omega_n)$ of algebra weights on $\mathbb R^+$ we study various properties of the Fr\'{e}chet algebra $A(\omega)=\bigcap_n L^1(\omega_n)$ obtained as the intersection of the weighted Banach algebras $L^1(\omega_n)$. We show that every endomorphism of $A(\omega)$ is standard, if for all n\in\mathbb N$there exists$m\in\mathbb N$such that$\omega_m(t)/\omega_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*\omega_n(t)/\omega_m(t)$is bounded on$\mathbb R^+$, then the continuous derivations on$A(\omega)$are exactly the linear maps$D$of the form$D(f)=(Xf)*\mu$for$f\in A(\omega)$, where$\mu$is a measure in$B(\omega)=\bigcap_n M(\omega_n)$and$(Xf)(t)=tf(t)$for$t\in\mathbb R^+$and$f\in A(\omega)$. If the condition is not satisfied, we show that$A(\omega)$ has no non-zero derivations.