Spectral theory of functions in studying partial differential operators

Abstract

The work is devoted to studying the spectral properties of differential operators with constant coefficients defined on subspaces of bounded continuous functions. Our main methods are spectral theory of Banach modules, theory of functions, abstract harmonic analysis and theory of representations, which were developed and described in detail in the monograph by A. G. Baskakov “Harmonic Analysis in Banach Modules and the Spectral Theory of Linear Operators”, Voronezh, VSU Publ., 2016. We introduce the algebra of polynomials by means of which we define differential operators. We also introduce closed subspaces of the space of bounded continuous functions called homogeneous function spaces, which play an important role in the analysis. An important class of spectrally homogeneous spaces is introduced as well. We obtain results relating the zero set of a polynomial with the properties of kernels and images of differential operators defined by these polynomials. We define the notion of a regular at infinity polynomial (ellipticity-type conditions) and we provide important examples of partial differential operators defined by such polynomials. The conditions of invertibility of such differential operators are obtained. In particular, we obtain criteria of invertibility in spectrally homogeneous spaces and spaces of periodic function. We get a result on coincidence of the spectrum of a differential operator with the image of polynomial generating this operator in spectrally homogeneous spaces. Conditions of compactness of the resolvent of partial differential operators defined by polynomials regular at the infinity are found.