Direct and inverse problems for differential systems connected with Dirac systems and related factorization problems Damir ArovHarry Dym 34A5534B2046E2247B32canonical systemsinverse monodromy probleminverse spectral probleminverse input impedance and input scatteringDirac and Krein systemspositive semidefinite $J$-unitary Hamiltonians 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. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2662 10.1512/iumj.2005.54.2662 en Indiana Univ. Math. J. 54 (2005) 1769 - 1816 state-of-the-art mathematics http://iumj.org/access/