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The purpose of this paper is to compute the asymptotics of determinants offinite sections of operators that are trace class perturbations of Toeplitzoperators. For example, we consider the asymptotics in the case where thematrices are of the form $ (a_{i-j} \pm a_{i+j+1-k})_{i,j=0... N-1} $ with $k$is fixed. We will show that this example as well as some general classes ofoperators have expansions that are similar to those that appear in the StrongSzeg\"{o} Limit Theorem. We also obtain exact identitities for some of thedeterminants that are analogous to the one derived independently by Geronimoand Case and by Borodin and Okounkov for finite Toeplitz matrices. Theseproblems were motivated by considering certain statistical quantities thatappear in random matrix theory.

Determinant computations for some classes of Toeplitz-Hankel matrices