REPRESENTATION OF ATOMIC OPERATORS AND EXTENSION PROBLEMS

Abstract

The notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiiˇ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some $\sigma$-algebra to a larger space of functions measurable with respect to a larger $\sigma$-algebra, as well as to the possibility of extending any $\sigma$-homomorphism from a smaller-measure algebra to a $\sigma$-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective $\sigma$-algebras which provides such possibilities and induces the positive answer to the above representation problem.

AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05