A class of weighted convolution Fr\'echet algebras

TL;DR

This paper investigates the structure and properties of a class of weighted convolution Fréchet algebras formed by intersecting weighted $L^1$ spaces, focusing on endomorphisms and derivations under specific growth conditions.

Contribution

It characterizes the endomorphisms and continuous derivations of these algebras, providing conditions for their existence and explicit forms, advancing understanding of their algebraic structure.

Findings

Every endomorphism is standard under certain growth conditions.

Continuous derivations are characterized as convolution with measures when conditions are met.

No non-zero derivations exist if the growth conditions are not satisfied.

Abstract

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.