ON COMPLETENESS OF ROOT VECTORS OF THE CAUCHY PROBLEM OF THE FIRST ORDER EQUATION WITH DEVIATING ARGUMENT
Ключевые слова:
Spectrum, deviating argument, root subspace, completeness, Keldysh theorem, compactness, Hilbert-Schmidt theorem, Green function, resolvent, sheaf of Keldysh operators.Аннотация
The main objective of this paper is to study the conditions under which the system of finite-dimensional invariant (root) subspaces of the operator A turns out to be complete in H or in the range of the operator.
In the case of a general completely continuous operator, completeness may not take place. In the present work, by the method of M.V. Keldysh, the completeness of the system of root vectors of the operator corresponding to the Cauchy problem of a first-order equation with a variable coefficient and a deviating argument is proved. The Volterra operator of the ordinary differential equation corresponding to the Cauchy problem is well known, so the result is in sharp contrast with the known facts. It can be expected that the results obtained will find application in theoretical physics, the theory of signal transmission, especially in fiber optic communications. Since the coefficient of the equation is not assumed to be real, the corresponding operator is not self-adjoint; therefore, questions of basicity were not considered. The result obtained is formulated in terms of the coefficient of the equation, and is close to the necessary, which is confirmed by the constructed example. This condition is a consequence of the method used; perhaps with other methods they can take a different look.
Note that the square of the operator A generates a sheaf of operators in the space therefore, the results of the paper are of interest also for the theory of sheaf.
