Spectral theory of differential and weighted composition operators

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The main theoretical component of this project is to study Fredholm differential operators with unbounded operator coefficients and their Fredholm index in terms of dynamical properties of linear differential equations on Banach spaces. This includes an operator-theoretical treatment of Evans function as a Fredholm determinant. New methods will be developed in spectral analysis of weighted composition (evolution) semigroups and related abstract differential operators on Sobolev and Lebesgue spaces of Banach space-valued functions. Also, we will study connections between the weighted composition operators and associated differential operators, and topics such as maximal regularity of solutions of evolution problems, Atiyah-Patodi-Singer Spectral Flow and Index Theorem, infinite dimensional Morse Theory, Krein-Birman spectral shift function, and Fredholm determinants for operators with semi-separable kernels. The main applied component of this project is to use these methods to study the point and essential spectrum of the classical linearized Euler operator in dimensions two and three obtained by linearizing the Euler equations of hydrodynamics about a steady state.This project lies at the intersection of the theory of complex dynamical systems, the mathematical theory of equations containing many parameters, and stability theory of hydrodynamics describing the behavior of a fluid in equilibria. We will develop new methods of study of the weighted composition semigroups, the mathematical models describing various effects of transport along trajectories of the given complex system. These methods will be applied to the study of a linear approximation of the Euler equations governing the motion of an ideal incompressible fluid in dimensions two and three, which will add to our understanding of fluid dynamics near equilibria.