Asymptotics of Toeplitz matrices with symbols in some generalized Krein algebras

Citation:
Karlovich, Alexei Yu. "Asymptotics of Toeplitz matrices with symbols in some generalized Krein algebras." Modern Analysis and Applications: Mark Krein Centenary Conference, Vol. 1. Operator Theory Advances and Applications, 190. Eds. V. Adamyan, Y. Berezansky, I. Gohberg, M. Gorbachuk, V. Gorbachuk, A. Kochubei, H. Langer, and G. Popov. Basel: Birkhäuser, 2009. 341-359.

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 Szegö 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 Szegö-Widom asymptotic trace formula holds with $$o(1)$$ replaced by $$o(n^{1-\alpha-\beta})$$.