Octonionic isometric isomorphisms and partial isometry

TL;DR

This paper advances octonionic functional analysis by exploring properties of para-linear isometric operators, establishing their characterizations, and introducing partial isometries, with implications for octonionic Stiefel spaces.

Contribution

It introduces and characterizes octonionic para-linear isometric and partial isometric operators, linking them to orthonormal bases and extending the theory of octonionic Hilbert modules.

Findings

An octonionic matrix is an isometry iff its row vectors form a weak associative orthonormal basis.

Para-linear isometric operators map associative orthonormal bases to weak associative orthonormal bases.

The paper offers a new perspective on James questions via octonionic Stiefel space modifications.

Abstract

Very recently, two new notions of para-linear mappings and weak associative orthonormal bases were introduced in octonionic functional analysis, which have been proved to be powerful in formulating the basic theory, such as the Riesz representation theorem and the Parseval theorem. In this article, we continue exploring more properties of these two concepts and initiate the study of octonionic para-linear isometric operators. Surprisingly, it is proven that the condition of the para-linear operator on a Hilbert octonionic bimodule being an isometric isomorphism is equivalent to it mapping any associative orthonormal basis to a weak associative orthonormal basis, which implies also that an octonionic matrix is an isometry if and only if the system of its row vectors is a weak associative orthonormal basis. Furthermore, we introduce the concept of para-linear partial isometric operators and establish the aforementioned analogue in this new setting. Based on these facts, we can provide naturally a new viewpoint of James questions by modifying the definition of octonionic Stiefel space.