  A real semi-inner-product space is a real vector space $\M$ equipped with a
function $[.,.] : \M \times \M \to \Re$ which is linear in its first variable,
strictly positive and satisfies the Schwartz inequality. It is well-known that
the function $||x|| = \sqrt{[x,x]}$ defines a norm on $\M$. and vica versa, for
every norm on $X$ there is a semi-inner-product satisfying this equality. A
linear operator $A$ on $\M$ is called \emph{adjoint abelian with respect to
$[.,.]$}, if it satisfies $[Ax,y]=[x,Ay]$ for every $x,y \in \M$. The aim of
this paper is to characterize the diagonalizable adjoint abelian operators in
finite dimensional real semi-inner-product spaces satisfying a certain
smoothness condition.
