  We examine spectral operator-theoretic properties of linear and nonlinear
dynamical systems with globally stable attractors. Using the Kato Decomposition
we develop a spectral expansion for general linear autonomous dynamical systems
with analytic observables, and define the notion of generalized eigenfunctions
of the associated Koopman operator. We interpret stable, unstable and center
subspaces in terms of zero level sets of generalized eigenfunctions. We then
utilize conjugacy properties of Koopman eigenfunctions and the new notion of
open eigenfunctions - defined on subsets of state space - to extend these
results to nonlinear dynamical systems with an equilibrium. We provide a
characterization of (global) center manifolds, center-stable and
center-unstable manifolds in terms of joint zero level sets of families of
Koopman operator eigenfunctions associated with the nonlinear system.After
defining a new class of Hilbert spaces, that capture the on and off attractor
properties of dissipative dynamics, and introduce the concept of Modulated Fock
Spaces}, we develop spectral expansions for a class of dynamical systems
possessing globally stable limit cycles and limit tori, with observables that
are square-integrable in on-attractor variables and analytic in off-attractor
variables. We discuss definitions of stable, unstable and global center
manifolds in such nonlinear systems with (quasi)-periodic attractors in terms
of zero level sets of Koopman operator eigenfunctions. We define the notion of
isostables for a general class of nonlinear systems. We provide a simple
example of a measure-preserving system that is not chaotic but has continuous
spectrum, and discuss experimental observations of spectrum on such systems. We
define the coherent principal dimension for a class of datasets based on the
lattice-type principal spectrum of the associated Koopman operator.
