@inbook {7481, title = {Asymptotics of Toeplitz matrices with symbols in some generalized Krein algebras}, booktitle = {Modern Analysis and Applications: Mark Krein Centenary Conference, Vol. 1. Operator Theory Advances and Applications, 190}, year = {2009}, pages = {341-359}, publisher = {Birkh{\"a}user}, organization = {Birkh{\"a}user}, address = {Basel}, abstract = {

Let \(\alpha,\beta\in(0,1)\) and
\[
K^{\alpha,\beta}:=\left\{a\in L^\infty(\mathbb{T}):\
\sum_{k=1}^\infty |\widehat{a}(-k)|^2 k^{2\alpha}\<\infty,\
\sum_{k=1}^\infty |\widehat{a}(k)|^2 k^{2\beta}\<\infty
\right\}.
\]
Mark Krein proved in 1966 that \(K^{1/2,1/2}\) forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended
earlier results of Gabor Szeg{\"o} for scalar symbols and established the asymptotic trace formula
\[
\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1)
\quad\text{as}\ n\to\infty
\]
for finite Toeplitz matrices \(T_n(a)\) with matrix symbols \(a\in K^{1/2,1/2}_{N\times N}\). We show that if \(\alpha+\beta\ge 1\) and \(a\in K^{\alpha,\beta}_{N\times N}\), then the Szeg{\"o}-Widom asymptotic trace formula holds with \(o(1)\) replaced by \(o(n^{1-\alpha-\beta})\).

}, url = {http://link.springer.com/chapter/10.1007/978-3-7643-9919-1_21}, author = {Karlovich, Alexei Yu.}, editor = {Adamyan, V and Berezansky, Y and Gohberg, I and Gorbachuk, M and Gorbachuk, V and Kochubei, A and Langer, H and Popov, G} }