Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

Let $\alpha ,\beta \in (0,1)$ and $$K^{\alpha ,\beta }:=\{a\in L^{\mathrm{\infty }}(\text{T}):\sum _{k=1}^{\mathrm{\infty }}\mathrm{\mid }\widehat{a}(-k){\mathrm{\mid }}^2{k}^{2\alpha }<\mathrm{\infty },\sum _{k=1}^{\mathrm{\infty }}\mathrm{\mid }\widehat{a}(k){\mathrm{\mid }}^2{k}^{2\beta }<\mathrm{\infty }\}\mathrm{.}$$ Mark Krein proved in 1966 that $K^{1\mathrm{/}2,1\mathrm{/}2}$ 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\H{o} for scalar symbols and established the asymptotic trace formula $$\mathrm{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1)\text{as }n\to \mathrm{\infty }$$ for finite Toeplitz matrices $T_n(a)$ with matrix symbols $a\in K^{1/2,1/2}{N\times N}$\mathrm{.}Weshowthatif$\alpha+\beta\ge 1$and$a\in K^{\alpha,\beta}{N\times N}$, then the Szeg\H{o}-Widom asymptotic trace formula holds with $o(1)$ replaced by $o(n^{1-\alpha -\beta })$.