UC Santa Cruz

Let $A=(a_{j,k})_{j,k=-\infty}^\infty$ be a bounded linear operator on $l^2(\Z)$ whose diagonals

$D_n(A)=(a_{j,j+n})_{j=-\infty}^\infty \in l^\infty(\Z)$ 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

$A_{n_1,n_2}=(a_{j,k})_{j,k=n_1}^{n_2-1}$ as their size $n_2-n_1$ tends to infinity.

Examples of such operators include block Toeplitz operators and the almost Mathieu operator.