Properties of derivations on some convolution algebras

TL;DR

This paper characterizes derivations on certain convolution algebras, showing they are represented by measures and analyzing their compactness and continuity properties.

Contribution

It provides a complete description of derivations on specific convolution algebras and examines their compactness, Montel properties, and dual space extensions.

Findings

Derivations are of the form D_μ f = Xf * μ with measures μ.

Characterization of weakly compact and Montel derivations.

Extension of derivations to dual spaces is weak-star continuous.

Abstract

For all the convolution algebras $L^1[0,1),\ L^1_{\text{loc}}$ and $A(\omega)=\bigcap_n L^1(\omega_n)$, the derivations are of the form $D_{\mu} f=Xf*\mu$ for suitable measures $\mu$, where $(Xf)(t)=tf(t)$. We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure $\mu$. Moreover, for all these algebras we show that the extension of $D_{\mu}$ to a natural dual space is weak-star continuous.