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This paper extends Yosida's mean ergodic theorem in order to computeprojections onto non-unitary eigenspaces for spectral operators of scalar-typeon locally convex linear topological spaces. For spectral operators withdominating point spectrum, the projections take the form of Laplace averages,which are a generalization of the Fourier averages used when the spectrum isunitary. Inverse iteration and Laplace averages project onto eigenspaces ofspectral operators with minimal point spectrum. Two classes of dynamicalsystems --- attracting fixed points in $\mathbb{C}^{d}$ and attracting limitcycles in $\mathbb{R}^{2}$ --- and their respective spaces of observables aregiven for which the associated composition operator is spectral. It is shownthat the natural spaces of observables are completions with an $\ell^{2}$polynomial norm of a space of polynomials over a normed unital commutativering. These spaces are generalizations of the Hardy spaces $H^{2}(\mathbb{D})$and $H^{2}(\mathbb{D}^{d})$. Elements of the ring are observables defined onthe attractor --- the fixed point or the limit cycle, in our examples.Furthermore, we are able to provide a (semi)global spectral theorem for thecomposition operators associated with a large class of dissipative nonlineardynamical systems; any sufficiently smooth dynamical system topologicallyconjugate to either of the two cases above admits an observable space on whichthe associated Koopman operator is spectral. It is conjectured that this isgenerically true for systems where the basin of attraction can be properly"coordinatized".

Construction of eigenfunctions for scalar-type operators via Laplace averages with connections to the Koopman operator