Groups of isometries on Orlicz spaces

Abstract

1* Introduction* Let X be a real or complex Orlicz space of functions on an atomic measure space; an additional (not very restrictive) condition will be imposed on X which implies in particular that X Φ L°°. If X is a Hubert space, there are numerous strongly continuous one parameter groups of isometries on X, according to a classical theorem of M. H. Stone; namely, each skew-adjoint operator on X generates such a group. We shall show that this property characterizes the Hubert spaces among the Orlicz spaces under consideration on an atomic measure space. Our main result is, roughly, if {Tt: teR ~ (—°°, °°)} is a strongly continuous (or (Co)) group of linear isometries on X and if X is not a Hubert space, then for each real t, Tt has the following form: (Ttf)(w) = exp {i tg(w)}f(w) for fe X and w e Ω if X is complex, where g is a real-valued function on Ω; or Tt = I (= the identity operator on X) if X is real. Section 2 contains some preliminaries, including a discussion of duality maps for Orlicz spaces. The main result is stated and proved in § 3. Section 4 contains some complements and examples, including a proof of the main theorem for finite dimensional L°° spaces. The present paper has several points of contact with Lumer's paper [9], which we became aware of shortly after the present paper was submitted for publication.