Regularity of bounded tri-linear maps and the fourth adjiont of a tri-derivation

TL;DR

This paper establishes a criterion for the regularity of bounded tri-linear maps and explores conditions under which the fourth adjoint of a tri-derivation remains a tri-derivation.

Contribution

It provides a simple regularity criterion for tri-linear maps and characterizes when the fourth adjoint of a tri-derivation is also a tri-derivation.

Findings

Regularity of tri-linear maps characterized by composition with linear maps.

Necessary and sufficient conditions for the fourth adjoint of a tri-derivation to be a tri-derivation.

Simplified criteria for regularity and adjoint properties in tri-linear mappings.

Abstract

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is a bounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linear mapping, then $f$ is regular if and only if $hof$ is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint $D^{****}$ of a tri-derivation $D$ is again tri-derivation.