M.Sc Thesis
M.Sc Student Igor Margulis Inertia Theorems Based on Operator Lyapunov Equations and their Applications Department of Mathematics Professor Emeritus Lerer Leonid (Deceased)

Abstract

Operator equations of Lyapunov type play an important role in various areas of pure and applied mathematics and in theoretical engineering. Lyapunov equations are closely related to the problem of determining the localization of spectrum of matrix or linear operator. The theory which studies how the spectra of two matrices or linear operators, satisfying Lyapunov equation, are related is called the inertia theory.

While the problem is completely solved for such equations with semi definite right hand side in the finite dimensional case, not too much is known in this case for the infinite dimensional operators. In this work we present new inertia results in the infinite dimensional setting, which are significant improvement on known results.

Both bounded and unbounded operators are considered.

The proof of these new inertia results is obtained using a new original approach to the problem and some sophisticated methods in the theory of operators acting in infinite dimensional spaces endowed with indefinite inner product.

In this work we present also new solutions for well-known direct and inverse problems for Krein continuous orthogonal functions. We develop a completely new approach which is based on representing the Krein functions as realizations of curtain infinite dimensional, time-invariant linear systems. The crucial point of this approach is that the associated main operator associated with Krein continuous orthogonal function satisfies Lyapunov equation with semi definite right hand side. Thus using our inertia results, we solve the direct problem which studies the distribution of zeros of Krein continuous orthogonal function. Using the Lyapunov equations we find also the solution for the inverse problem, i.e. given an entire function of specific form we find the integral operator for which this function is a Krein continuous orthogonal function. The explicit formula for the integral operator is new and did not exist in earlier works on the topic.

Besides applying the results, we obtain for Lyapunov equations, to Krein orthogonal functions, we examine our inertia results in other areas. In particular, we study the connection with the control theory and semigroups of linear operators.