Isometries of certain function spaces

Let X be a discrete symmetric Banach function space with absolutely continuous norm. We prove by the method of generalized hermitian operator that an operator U on X is an onto isometry if and only if it is of the form: Uf(.)=u(.)f(T.) all feX, where % is a unimodular function and T is a set isomorphism of the underlying measure space. That other types of isometries occur if the symmetry condition is not present is illustrated by an example. We completely describe the isometries of a reflexive Orlicz space LMΦ(Γ^LZ) provided the atoms have equal mass (the atom-free case has been treated by G. Lumer); similarly for the case that no Hubert subspace occurs.