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 $$⟨f(x),g(\alpha )⟩=⟨x,\alpha ⟩\forall x\in E,\forall \alpha \in E^{\ast },$$ where $f:E\to F,g:E^{\ast }\to F^{\ast }$ are two maps. Our result generalizes the recent result of Łukasik and Wójcik in the case that $E$ and $F$ are Hilbert spaces.

The proof of Lemmas 1 and 2 was added. Keywords: orthogonality equation; Banach space; bounded linear operator