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 f ∈ X, where u 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(≠L2) provided the atoms have equal mass (the atom-free case has
been treated by G. Lumer); similarly for the case that no Hilbert subspace
occurs.
