Operator Pencils and Half-range Problem in Operator Theory

TL;DR

This paper introduces operator pencils and half-range problems in operator theory, focusing on wave selection, factorization of operator pencils, and applications to mechanics, highlighting differences between finite and infinite dimensional cases.

Contribution

It presents new methods for factorizing operator pencils and applying abstract models to concrete mechanical problems, including stability criteria using Pontrjagin space techniques.

Findings

Complete factorization solutions in finite dimensions

Factorization of elliptic operator pencils with resolvent growth condition

Stability criteria for Sobolev problem using Pontrjagin space methods

Abstract

This article can be considered as the first version of a book which the author plans to write about half-range problems in operator theory. It consists of two parts. The first part is based on lectures which the author delivered at University of Calgary and Lomonosov Moscow State University. The main attention in this part is paid to the selection of waves which are involved in the formulation of the Mandelstamm radiation principle (the eigen-pairs, corresponding to the real eigenvalues) and to the factorization problems of self-adjoint and dissipative, quadratic and polynomial operator pencils. There is a dramatic difference between finite dimensional and infinite dimensional cases. It is shown that in the finite dimensional case the factorization problems can be solved completely. In the second part we consider abstract models for concrete problems of mechanics. We demonstrate the methods how concrete problems can be represented in an abstract form. The main results concern the factorization of elliptic operator pencils satisfying the resolvent growth condition in a double sector containing the real axis and the investigation of the semi-group properties of a divisor. Using Pontrjagin space methods we obtain a criterium for the stability in the celebrated Sobolev problem about a rotating top with a cavity filled with a viscous liquid.