Dalets'kyj O. Differential and pseudodifferential operators on infinite dimensional manifolds

The thesis is devoted to the development of the theory of differential and pseudodifferential operators on infinite dimensional manifolds. The main attention is paid to the probabilistic approach of study of second order elliptic operators on product-manifolds (infinite products of compact Riemannian manifolds) and to asymptotic methods of investigation of pseudodifferential operators with Hilbert (and Hilbert manifold) phase spaces. In particular, Feller semigroups associated with Dirichlet operators of Gibbs measures on product-manifolds are constructed. The symbolic calculus for an important class of pseudodifferential operators with Hilbert phase space and semiclassical solutions of corresponding Schroedinger equations are constructed. The asymptotic quatntization functor for a class of infinite dimensional symplectic manifolds is constructed.