Conference - DimaScat : Scattering Theory and Spectral Asymptotics of Differential Operators - in Honour of Dimitri Yafaev

We present a setting generalizing that for Szegö's theorem on asymptotics of Toeplitz determinants. The setting is given in terms of two functions (a test function and a symbol) and an underlying ergodic operator in $l^2(\mathbb{Z}^d)$, in particular, the Schrödinger operator with ergodic potential. The setting leads to a variety of asymptotic trace formulas determined by the smoothness of the two functions and mixing properties of the coefficients of ergodic operator. We discuss some of the formulas including those for the large block asymptotics of entanglement entropy of certain bipartite disordered quantum systems.