A Finer Aspect of Eigenvalue Distribution of Selfadjoint Band Toeplitz Matrices

The asymptotics of eigenvalues of Toeplitz operators has received a lot of attention in the mathematical literature and has been applied in several disciplines. This paper describes two such application disciplines and provides refinements of existing asymptotic results using new methods of proof. The following result is typical: Let $T(\varphi)$ be a selfadjoint band limited Toeplitz operator with a (real valued) symbol $\varphi$, which is a nonconstant trigonometric polynomial. Consider finite truncations $T_n(\varphi)$ of $T(\varphi)$, and a finite union of finite intervals of real numbers $E$. We prove a refinement of the Szego asymptotic formula \[ \lim_{n\rightarrow \infty }\frac{N_n(E)}n=\frac 1{2 \pi}m(F). \] Indeed, we show that \[ N_n(E) - \frac 1{2\pi}m(F)n = {\it O}(1). \] Here $m(F)$ denotes the measure of $F = \varphi^{-1}(E)$ on the unit circle, and $N_n(E)$ denotes the number of eigenvalues of $T_n(\varphi)$ inside $E$. We prove similar results for singular values of general Toeplitz operators involving a refinement of the Avram--Parter theorem.