Decomposition of functions between Banach spaces in the orthogonality equation

TL;DR

This paper characterizes solutions to a Banach-orthogonality equation involving Banach spaces, extending previous Hilbert space results to the reflexive Banach space setting.

Contribution

It generalizes recent solutions of the orthogonality equation from Hilbert spaces to reflexive Banach spaces, providing a broader understanding.

Findings

Provides a description of solutions when F is reflexive

Extends Hilbert space results to Banach spaces

Generalizes previous work by Lukasik and Wójcik

Abstract

Let $E,F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f,g)$ of the Banach-orthogonality equation $$\langle f(x),g(\alpha)\rangle=\langle x,\alpha\rangle\hspace{10mm}\forall x\in E,\forall \alpha\in E^*,$$ where $f:E\rightarrow F,g:E^*\rightarrow F^*$ are two maps. Our result generalizes the recent result of {\L}ukasik and W\'{o}jcik in the case that $E$ and $F$ are Hilbert spaces.