Calculus of Variations and Geometric Measure Theory

E. Stepanov

Representation of atomic operators and extension problems

created on 01 Jul 2003
modified by stepanov on 22 Nov 2004


Published Paper

Inserted: 1 jul 2003
Last Updated: 22 nov 2004

Journal: Proc. Edunburgh Math. Soc.
Volume: 47
Pages: 695-707
Year: 2004


The notion of an atomic operator between spaces of measurable functions (M. Drakhlin, A. Ponosov, E. Stepanov. On some classes of operators determined by the structure of their memory, Proc. Edinburgh Math. Soc., 45(2002), no. 2, 467-490) provides 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 Nemytski\v{\i} (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 $\sigma$-algebrae which provides such possibilities and induces the positive anwer to the above representation problem.

Keywords: shift operator, composition operator, local operator, atomic operator, Nemytski\v{\i} operator, Carathéodory function, representation