Asymptotics of determinants for finite sections of operators with almost periodic diagonals

Let $A=(a_{j,k}{)}_{j,k=-\mathrm{\infty }}^{\mathrm{\infty }}$ be a bounded linear operator on $l^2(\mathbb{Z})$ whose diagonals $D_n(A) = (a_{j,j-n})_{j=-\infty}^\infty\in l^\infty($\mathbb{Z}$)$ are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants $\det A_{n_1,n_2}$ of the finite sections of the operator $A$ as their size $n_2-n_1$ tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.