Decomposition of functions between Banach spaces in the orthogonality equation

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

$$\begin{aligned} \langle f(x),g(\alpha )\rangle =\langle x,\alpha \rangle \qquad \forall x\in E,\forall \alpha \in E^*, \end{aligned}$$

where $f:E\rightarrow F,g:E^*\rightarrow F^*$ are two maps. Our result generalizes the recent result of Łukasik and Wójcik in the case that E and F are Hilbert spaces.