Infinite-Dimensional Elliptic Differential operators of the Second Order

Abstract

Infinite-dimensional differential operators appear in different branches of mathematics and their applications. Thus, in mathematical physics, they are used (though often on a formal level) as operators of energy of systems with infinitely many degrees of freedom; in the theory of random processes where diffusion processes with infinite-dimensional phase spaces are constructed with the help of operators of this sort; and, finally, the investigation of infinite-dimensional differential operators is of independent interest for infinite-dimensional analysis. These directions are not isolated, on the contrary, they interact intensively. However, each of these branches has its own specific circle of problems and, thus, if we choose this or another field of applications, then we automatically draw attention to some concrete aspects of the theory of infinite-dimensional differential operators which must be studied. For example, applications to theoretical physics are mainly based on operator problems whose contents have many similar features for different model situations and can be characterized by close connection with spectral theory. An essential place among these problems is occupied by the problems of selfadjointness and the theory of singular perturbations. These problems play a key role in both Hamiltonian and Euclidean approaches to the construction and investigation of the dynamics of many important model systems in contemporary mathematical physics. The necessity of the investigation of these problems largely determines the scope of problems in the theory of infinite-dimensional differential operators that are presented in this chapter.