On Musielak-orlicz Spaces Isometric to L 2 or L ∞

It is proved that a Musielak-Orlicz space LΦ of real valued functions which is isometric to a Hilbert space coincides with L2 up to a weight, that is Φ(u,t)= c(t)u2. Moreover it is shown that any surjective isometry between LΦ and L∞ is a weighted composition operator and a criterion for LΦ to be isometric to L∞ is presented. Isometries in complex function spaces have been studied successfully for fairly long time (see review article by Fleming and Jamison [3]). The real case appeared more difficult and since the well known characterization of isometries in Lp spaces by S. Banach in 1932 and some partial results in other spaces, only very recently such isometries in rearrangement invariant real function spaces have been characterized by Kalton and Randrianantoanina in [7]. Isometries between r.i. real spaces and Lp have been also studied in [2]. Here we study surjective linear isometries in the class of Musielak-Orlicz spaces of real-valued functions. We show that any MusielakOrlicz space LΦ isometric to a Hilbert space must coincide with L2 “up to a weight”, namely there exists a positive measurable function c(t), such that Φ(u, t) = c(t)u. We also present a criterion for LΦ to be isometric to L∞, and moreover we show that any such isometry has disjoint support property and in consequence is a weighted composition operator. In the sequel let (Ω,Σ, μ) be a σ-finite, complete and nonatomic measure space. Symbols R, R+, N and Z stand, as usual, for reals, nonnegative reals, natural numbers and for integers respectively. The space of all (equivalence classes of) Σmeasurable real functions defined on Ω is denoted by L. L is a lattice with the pointwise order, that is f ≤ g whenever f(t) ≤ g(t), a.e. As usual L2 and L∞ stand