Adjoint abelian operators on $L^{p}$ and $C(K)$

Abstract:

An operator A on a Banach space X is said to be adjoint abelian if there is a semi-inner product $[ \cdot , \cdot ]$ consistent with the norm on X such that $[Ax,y] = [x,Ay]$ for all $x,y \in X$. In this paper we show that every adjoint abelian operator on $C(K)$ or ${L^p}(\Omega ,\Sigma ,\mu )\;(1 < p < \infty ,p \ne 2)$ is a multiple of an isometry whose square is the identity and hence is of the form $Ax( \cdot ) = \lambda \alpha ( \cdot )(x \circ \phi )( \cdot )$ where $\alpha$ is a scalar valued function with $\alpha ( \cdot )\alpha \circ \phi ( \cdot ) = 1$ and $\phi$ is a homeomorphism of K (or a set isomorphism in case of ${L^p}(\Omega ,\Sigma ,\mu ))$ with $\phi \circ \phi =$ identity (essentially).