Injective isometries in Orlicz spaces

Details Injective isometries in Orlicz spaces Injective isometries in Orlicz spaces

1998-12-09

arxiv.org/abs/math/9812062v1

Mathematics

Functional Analysis

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

Scientific paper

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 $M''$, satisfies another ``smoothness type'' condition and either $\lim_{t\to0} M''(t) = \infty$ or $M''(0) = 0$ and $M''(t)>0$ 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.

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