Pareto vectors of continuous linear operators

Abstract

The intersection of all zero-neighborhoods in a topological module over a topological ring is a bounded and closed submodule whose inherited topology is the trivial topology. In this manuscript, we prove that this is the smallest closed submodule and thus replaces the null submodule in the Hausdorff setting. This fact motivates to introduce a new notion in operator theory called topological kernel. Another new concept is also defined that of Pareto optimal element for a family of continuous linear operators between topological modules. It is then proved that topological kernels have a strong influence on the existence of Pareto optimal elements. This work is strongly motivated by the ongoing search for a consistent operator theory on topological modules over general topological rings.