The spectral property of dual matrix strings as a consequence of the existence of spectral matrix functions for canonical systems

The term dual string for scalar strings was introduced in KK1], where some connections between the spectra of a string and its dual were studied. In KK2] it was shown that if () is a spectral function of a scalar stringS ₁ with nonnegative spectrum (in the sense of KK2]), then the function is a spectral function of the string (S _d)₀ which isfully dual toS ₁. This result was generalized to regular matrix strings with continuous invertible matrix densities by H. Dym and L. A. Sakhnovich DS]. In the present work we generalize in part the mentioned result from DS] to matrix strings that may be singular, and may have matrix density that is everywhere discontinuous and noninvertible on a set of positive measure.