*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

**Abstract:**

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

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