  Generalized indefinite strings provide a canonical model for self-adjoint
operators with simple spectrum (other classical models are Jacobi matrices,
Krein strings and 2x2 canonical systems). We prove a number of Szeg\H{o}-type
theorems for generalized indefinite strings and related spectral problems
(including Krein strings, canonical systems and Dirac operators). More
specifically, for several classes of coefficients (that can be regarded as
Hilbert-Schmidt perturbations of model problems), we provide a complete
characterization of the corresponding set of spectral measures. In particular,
our results also apply to the isospectral Lax operator for the conservative
Camassa-Holm flow and allow us to establish existence of global weak solutions
with various step-like initial conditions of low regularity via the inverse
spectral transform.
