Bezoutian and Schur-Cohn problem for operator polynomials (Q1059828)

A generalized Bezout operator (Bezoutian) for a pair of operator polynomials is introduced and its kernel is described in terms of common spectral data of the underlying polynomials. The location of the spectrum of an operator polynomial with compact spectrum with respect to the unit circle (infinite-dimensional version of the Schur-Cohn problem) is expressed via the inertia of a suitable Bezoutian. An application to the geometric dichotomy problem for difference equations with operator coefficients is given as well.