On time-dependent orthogonal polynomials on the unit circle

Two index formulas for operators defined by infinite band matrices are proved. These results may be interpreted as a generalization of a classical theorem of M. G. Krein on orthogonal polynomials. The proofs are based on dichotomy and nonstationary inertia theory.Dedicated to the memory of M. G. Krein, a mathematical giant, a great teacher and wonderful friend.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 18–36, January–February, 1994.