Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants

Suppose that where and , and the Toeplitz operator is invertible. Let be the determinant of the Toeplitz matrix where . Let be the orthogonal projection onto where ; set , let denote the Hankel operator associated to , and set for . For the Wiener-Hopf factorization where and , put , Theorem A.    

Let be a decomposition into invariant subspaces, and , so that restricted to is invertible, is finite dimensional, and restricted to is nilpotent. Let be the basis for the null space of , and let be the top vector in a Jordan root vector chain of length lying over , i.e., where . Theorem B.     , the holonomy of a Deligne bundle with connection defined by the factorization . Note that the generalizations of the Szegö limit theorem for which have appeared in the literature with instead of have the defect that the limit of does not exist in general. An example is given with yet for infinitely many .