Injective isometries in Orlicz spaces

Abstract

We show that injective isometries in Orlicz space $L_M$ have to preserve disjointness, provided that Orlicz function $M$ satisfies ${\Delta }_2$-condition, has a continuous second derivative $M^{\prime \prime }$, satisfies another ``smoothness type'' condition and either ${\lim }_{t\to 0}{M}^{\prime \prime }(t)=\infty$ or $M^{\prime \prime }(0)=0$ and $M^{\prime \prime }(t)>0$ for all $t>0$. The fact that surjective isometries of any rearrangement-invariant function space have to preserve disjointness has been determined before. However dropping the assumption of surjectivity invalidates the general method. In this paper we use a differential technique.

20 pages, 2 figures, to appear in the Proceedings of the Third Conference on Function Spaces held in Edwardsville in May 1998, Contemporary Math