REPRESENTATION OF ATOMIC OPERATORS AND EXTENSION PROBLEMS

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

BOUNDARY-CHAOTIC BEHAVIOUR OF CONTINUOUS FUNCTIONS UNDER THE ACTION OF OPERATORS

In this paper we introduce two classes of operators on spaces of continuous functions with values in $F$-spaces under the action of which many functions behave chaotically near the boundary. Several examples—including onto linear operators, left and right composition operators, multiplication operators, and operators with pointwise dense range or with some stability property—are given. This new theory extends one recently developed on spaces of holomorphic functions.AMS 2000 Mathematics subject classification: Primary 47B38. Secondary 30D40; 46E10; 54D45

ON SOME CLASSES OF OPERATORS DETERMINED BY THE STRUCTURE OF THEIR MEMORY

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 non-compactness in measure and weak degeneracy, while having 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 this paper.AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05