Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

Abstract

Let $α,β\in(0,1)$ and \[ K^{α,β}:=\left\{a\in L^\infty(\T): \sum_{k=1}^\infty |\hat{a}(-k)|^2 k^{2α}<\infty, \sum_{k=1}^\infty |\hat{a}(k)|^2 k^{2β}<\infty \right\}. \] 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 Szegő 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 $α+β\ge 1$ and $a\in K^{α,β}_{N\times N}$, then the Szegő-Widom asymptotic trace formula holds with $o(1)$ replaced by $o(n^{1-α-β})$.