Hello,

Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$.

If $A_n$ were a sequence of Hermitian Toeplitz matrices, and if $\sum_k|a_k|<\infty$, then Szego theorem states that for any continues function $F(\cdot)$ on $[\alpha,\beta]$ we have
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}F(\lambda_{n,k}) = \frac{1}{2\pi}\int_0^{2\pi}F(f(\xi))d\xi
$$

where
$$
f(\xi) = \sum_{k = -\infty}^{\infty}a_ke^{ik\xi}
$$
and $\alpha = \text{ess}\inf f$ and $\beta = \text{ess} \sup f$.

If however $A_n$ are not-Hermitian, then the above hold only for polynomial functions, namely $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}p(\lambda_{n,k}) = \frac{1}{2\pi}\int_0^{2\pi}p(f(\xi))d\xi $$ where $p(\cdot)$ is some polynomial function.

My question: is there any result regard the asymptotic behavior of the modulus of the eigenvalues for the non-Hermitian case, namely $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}|\lambda_{n,k}| = ?. $$

EDITION: due to the thankful Alexandersson comment, I add the factor $1/n$...