# On the Analogs of Szego Theorem for Ergodic Operators

# On the Analogs of Szego Theorem for Ergodic Operators

We consider an asymptotic setting for ergodic operators generalizing that for the Szego theorem on the asymptotics of determinants of finite-dimensional restrictions of the Toeplitz operators. The setting is formulated via a triple consisting of an ergodic operator and two functions, the symbol and the test function. We analyze in the frameworks of this setting the two important examples of ergodic operators: the one dimensional discrete Schrodinger operator with random i.i.d. potential and the same operator with quasiperiodic potential. In the first case we find that the corresponding asymptotic formula contains a new subleading term proportional to the square root of the length of the interval of restriction. The origin of the term are the Gaussian fluctuations of the corresponding trace, i.e, in fact, the Central Limit Theorem for the trace. In the second (quasiperiodic) case the subleading term is bounded as in the Szego theorem, but, unlike the theorem, where the term does not depend on the length, in the quasiperiodic case the term is an ergodic process in the length.