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We examine spectral operator-theoretic properties of linear and nonlineardynamical systems with globally stable attractors. Using the Kato Decompositionwe develop a spectral expansion for general linear autonomous dynamical systemswith analytic observables, and define the notion of generalized eigenfunctionsof the associated Koopman operator. We interpret stable, unstable and centersubspaces in terms of zero level sets of generalized eigenfunctions. We thenutilize conjugacy properties of Koopman eigenfunctions and the new notion ofopen eigenfunctions - defined on subsets of state space - to extend theseresults to nonlinear dynamical systems with an equilibrium. We provide acharacterization of (global) center manifolds, center-stable andcenter-unstable manifolds in terms of joint zero level sets of families ofKoopman operator eigenfunctions associated with the nonlinear system.Afterdefining a new class of Hilbert spaces, that capture the on and off attractorproperties of dissipative dynamics, and introduce the concept of Modulated FockSpaces}, we develop spectral expansions for a class of dynamical systemspossessing globally stable limit cycles and limit tori, with observables thatare square-integrable in on-attractor variables and analytic in off-attractorvariables. We discuss definitions of stable, unstable and global centermanifolds in such nonlinear systems with (quasi)-periodic attractors in termsof zero level sets of Koopman operator eigenfunctions. We define the notion ofisostables for a general class of nonlinear systems. We provide a simpleexample of a measure-preserving system that is not chaotic but has continuousspectrum, and discuss experimental observations of spectrum on such systems. Wedefine the coherent principal dimension for a class of datasets based on thelattice-type principal spectrum of the associated Koopman operator.

Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State Space Geometry