Calculus of Variations and Geometric Measure Theory

M. Drakhlin - A. Ponosov - E. Stepanov

On some classes of operators determined by the structure of their memory

created on 05 Feb 2002
modified on 01 Oct 2002


Published Paper

Inserted: 5 feb 2002
Last Updated: 1 oct 2002

Journal: Proc. Edinburgh Math. Soc.
Volume: 45
Number: 2
Pages: 467-490
Year: 2002


Two classes of nonlinear operators generalizing the notion of a local operator between ideal function spaces are introduced. The first class, called atomic, contains in particular all the linear shifts, while the second one, called coatomic, contains all the adjoints to former, and in particular, the conditional expectations. Both classes include local (in particular, Nemytski\v{\i}) operators and are closed with respect to compositions of operators. Basic properties of operators of introduced classes in the Lebesgue spaces of vector-valued functions are studied. It is shown that both classes inherit from Nemytski\v{\i} operators the properties of noncompactness in measure and weak degeneracy, while have different relationships of acting, continuity and boundedness, as well as different convergence properties. Representation results for the operators of both classes are provided. The definitions of the introduced classes as well as the proofs of their properties are based on a purely measure theoretic notion of memory of an operator, also introduced in the paper.

Keywords: local operator, Nemytski\v{i} operator, disjointness preserving operator, ideal function spaces, $\sigma$-homomorphism