My question is about computing asymptotic the determinant (dimension of the matrix $n\to\infty$) of a $2\times 2$ block Toeplitz matrix. $$\mbox{det}\left(\begin{array}{cc} a_n & b_n \\ d_n & e_n \end{array} \right)$$ where $a_n$, $b_n$, $c_n$ and $d_n$ are Toeplitz matrices. To illustrate the meaning of a Toeplitz matrix, you can have a look to the wikipedia article https://en.wikipedia.org/wiki/Toeplitz_matrix
I know that in the scalar case, the determinant can be computed by using the Szegö's theorems https://www.encyclopediaofmath.org/index.php/Szeg%C3%B6_limit_theorems \begin{equation} \lim_{n\to\infty}\mbox{det}(T_n(\alpha))=\left(G(\alpha)\right)^n E(\alpha) \end{equation} The dominant term at big dimension, $G$, can be easily extended to the matritial case as $G(\alpha)=\exp[(\log(\mbox{det}\;\alpha ))_0]$. However, I found no close expression for the correction $E(\alpha)$.
Is there any way to extend these theorems to the matritial case? I am interested on analytic or semi-analytic expressions. However, if it is not possible, numerical solutions, factorizations or simplifications are also welcomed. Thanks for any help!
Note: The symbol has in general no symmetries that would lead to a simplification of the problem, i.e. $$ T_n=\left(\begin{array}{cc} a_n & b_n \\ d_n & e_n \end{array} \right) $$ where $a_n$, $b_n$, $c_n$ and $d_n$ are in general independent.