Operator Theory. Nonclassical Problems Nonclassical Problems

This monograph deals with mathematical methods applicable to studying nonclassical problems of mathematical physics. Many problems of this type are reduced to equations, where the operators involved are noninvertible. In this case, the author uses special decompositions of an operator, generalized resolvents, semigroups with kernels, and some other approaches. The simplest model of this type is the first order operator-differential equation with a noninvertible operator in front of the derivative. The corresponding spectral problems arising here are the well-known problems for linear pencils. These and other problems are studied with the use of methods, which are based on the interpolation theory for Banach spaces. The emphasis is on applications of this theory to the theory of linear operators in indefinite inner product spaces, to studying the property of a linear operator to be exponentially dichotomous, to some continuity properties of linear operators in Hilbert scales, to the Riesz basisproperties of eigenelements and associated elements of linear pencils and the corresponding elliptic problems with an indefinite weight functions, and to studying nonclassical boundary value problems for first order operator-differential equations.Pyatkov, S. G. is the author of 'Operator Theory. Nonclassical Problems Nonclassical Problems' with ISBN 9789067643634 and ISBN 9067643637.