Differential Equations and Dynamics Seminar

Abstract: A classical result of harmonic analysis is that if the almost periodic spectrum Sp_{ap}(f) of a uniformly continuous bounded function f is countable, then f is an almost periodic function (Loomis Theorem). This result is an extension of the Wiener Tauberian Theorem. It has been extended to vector valued functions (with values in a Banach space), and has been applied to the study of properties of solutions of various classes of linear differential or integral equations.

In this talk, an abstract operator-analytic framework is proposed which, in particular, unifies these results for differential and integral equations. Namely, we consider an abstract operator equation of the form Du-Au=f, where D is the generator of a group of isometric operators on a Banach space E and A is a closable linear operator which commutes with D. We introduce the notion of equation spectrum, Sigma, associated with the pair A, D, and show that if u is a solution of the above equation, then Sp_{ap}(u) is a subset of Sigma intersect Sp_{ap}(f). This implies, that if f is an almost periodic element (w.r.t. the group generated by D) and Sigma is countable, then Sp_{ap}(u) also is countable. This implies almost periodicity of u, under some natural conditions.

Analogous approach is applicable to equations in Hilbert spaces, but the specifics, methods  and results are different, since harmonic analysis methods are now replaced by the spectral theorem.