IUMJ

Title: Direct and inverse problems for differential systems connected with Dirac systems and related factorization problems

Authors: Damir Z. Arov and Harry Dym

Issue: Volume 54 (2005), Issue 6, 1769-1816

Abstract:

Uniqueness theorems for inverse problems for canonical differential systems of the form $y'(t,\lambda) = i\lambda y(t,\lambda)H(t)J$ when $H(t) = X(t)NX(t)^{*}$ for appropriately restricted $X(t)$ and $N$ are established. These results are obtained by showing that the canonical differential systems under consideration can be imbedded into a general framework for which uniqueness theorems were obtained by the authors earlier. Subsequently, uniqueness theorems for a class of systems of the form $u'(t,\lambda) = i\lambda u(t,\lambda)NJ + u(t,\lambda)\mathcal{V}(t)$ that are related to Dirac systems are deduced on the basis of a factorization theorem that is developed in this paper. Finally, some refinements in a number of statements that are based on integral representations of matrix valued functions of the Schur class and the Caratheodory class are discussed briefly. A number of the basic observations in this last part were first noted by M.G. Krein.