Some properties of bounded tri-linear maps

TL;DR

This paper investigates properties of bounded tri-linear maps, introducing topological centers, exploring their relationships with weak compactness, regularity, and factorization, and establishing conditions for extensions and factorization.

Contribution

It defines topological centers for bounded tri-linear maps and analyzes their properties, linking regularity, weak compactness, and extension factorization in a novel way.

Findings

Introduces topological centers for tri-linear maps.

Establishes conditions for extension factorization.

Connects regularity with factorization properties.

Abstract

Let $X,Y,Z$ and $W$ be normed spaces and $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear mapping. In this Article, we define the topological centers for bounded tri-linear mapping and we invistagate thier properties. We study the relationships between weakly compactenss of bounded linear mappings and regularity of bounded tri-linear mappings. For both bounded tri-linear mappings $f$ and $g$, let $f$ factors through $g$, we present necessary and suficient condition such that the extensions of $f$ factors through extensions of $g$. Also we establish relations between regularity and factorization property of bounded tri-linear mappings.