We use the concept of a
wandering subspace to study isometries on spaces with an inner product that is not
assumed to be positive definite. The theory in many respects parallels the Hilbert
space theory, but there are significant differences that are emphasized here. Examples
are given which illustrate the complications that can arise when the inner product is
indefinite.
The first few sections of this paper are devoted to the study of indefinite inner
product spaces with admissible topologies, and the continuous operators on these
spaces. The rest of the paper concentrates on isometric operators, their wandering
subspaces, and the Fourier representations of shifts.
