During the investigation of large systems described by evolution equations, we encounter many problems. Of special interest is the problem of "high dimensionality" or, more precisely, the problem of the complexity of the phase space. The notion of the "comple xity of the. phase space" includes not only the high dimensionality of, say, a system of linear equations which appear in the mathematical model of the system (in the case when the phase space of the model is finite but very large), as this is usually understood, but also the structure of the phase space itself, which can be a finite, countable, continual, or, in general, arbitrary set equipped with the structure of a measurable space. Certainly, 6 6 this does not mean that, for example, the space (R 6, ( ), where 6 is a a-algebra of Borel sets in R 6, considered as a phase space of, say, a six-dimensional Wiener process (see Gikhman and Skorokhod [1]), has a "complex structure". But this will be true if the 6 same space (R 6, ( ) is regarded as a phase space of an evolution system describing, for example, the motion of a particle with small mass in a viscous liquid (see Chandrasek har [1]).

This volume is devoted to theoretical results which formalize the concept of state lumping: the transformation of evolutions of systems having a complex (large) phase space to those having a simpler (small) phase space. The theory of phase lumping has aspects in common with averaging methods, projection formalism, stiff systems of differential equations, and other asymptotic theorems. Numerous examples are presented in this book from the theory and applications of random processes, and statistical and quantum mechanics which illustrate the potential capabilities of the theory developed. The volume contains seven chapters. Chapter 1 presents an exposition of the basic notions of the theory of linear operators. Chapter 2 discusses aspects of the theory of semigroups of operators and Markov processes which have relevance to what follows. In Chapters 3--5, invertibly reducible operators perturbed on the spectrum are investigated, and the theory of singularly perturbed semigroups of operators is developed assuming that the perturbation is subordinated to the perturbed operator. The case of arbitrary perturbation is also considered, and the results are presented in the form of limit theorems and asymptotic expansions. Chapters 6 and 7 describe various applications of the method of phase lumping to Markov and semi-Markov processes, dynamical systems, quantum mechanics, etc. The applications discussed are by no means exhaustive and this book points the way to many more fruitful applications in various other areas. For researchers whose work involves functional analysis, semigroup theory, Markov processes and probability theory.