Self-adjoint operators on real Banach spaces

The aim of this paper is to discuss a functional equation ρ+′(f(x),y)=ρ+′(x,f(y)) for all x,y∈Xx,y∈X. We show that, if a mapping f:X→Xf:X→X satisfies this functional equation, then ff must be a linear continuous operator and we solve this equation in the case when X=C(M)X=C(M). Moreover, we give a new characterization of inner product spaces.