Pointwise Composition Research Articles

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions (for example, when the terms of the terms in the sums in the range of the operator are restricted to the interval [−1,1]) can be studied by adapting the classical analytical techniques of the summability of operators on Banach lattices, which recalls the work of Maurey. We show a technique to prove new integral dominations (equivalently, operator factorizations), which involve non-homogeneous expressions constructed by pointwise composition with Lipschitz maps. As an example, we prove a new family of integral bounds for certain operators on Lorentz spaces.

Endomorphism Seminear-Rings of Brandt Semigroups

We consider the endomorphisms of a Brandt semigroup B n and the semigroup of mappings E(B n ) that they generate under pointwise composition. We describe all the elements of this semigroup, determine Green's relations, consider certain special types of mapping, which we can enumerate for each n, and give complete calculations for the size of E(B n ) for small n.