  This paper considers a nonlinear dynamical system on a complex, finite
dimensional Banach space which has an asymptotically stable, hyperbolic fixed
point. We investigate the connection between the so-called principle
eigenfunctions of the Koopman operator and the existence of a topological
conjugacy between the nonlinear dynamics and its linearization in the
neighborhood of the fixed point. The principle eigenfunctions generate an
algebra of observables for the linear dynamics --- called the principle algebra
--- which can be used to generate a sequence of approximate conjugacy maps in
the same manner as it is done in normal form theory. Each element of the
principle algebra has an expansion into eigenfunctions of the Koopman operator
and composing an eigenfunction with one of the approximate topological
conjugacies gives an approximate eigenfunction of the Koopman operator
associated with the nonlinear dynamical system. When the limit of the
approximate conjugacies exists and attention is restricted to real Banach
spaces, a simple application of the Stone-Weierstrass theorem shows that both
the principle algebra and the pull-back algebra --- defined by composing the
principle algebra with the topological conjugacy --- are uniformly dense in
either the space of continuous functions or the maximal ideal of continuous
functions vanishing at the fixed point. The point is that, a priori, it is
difficult to know which space of observables to use for dissipative nonlinear
dynamical systems whose elements have spectral expansions into eigenfunctions.
These results say that any continuous observable is arbitrarily close to one
that has such an expansion.
