A new algebraic approach to the Szegő-Widom limit theorem

Abstract

We give another proof of the Szeg\H{o}–Widom Limit Theorem. This proof relies on a new Banach algebra method that can be directly applied to the asymptotic computation of the Toeplitz determinants. As a by-product, we establish an interesting identity for operator determinants of Toeplitz operators, namely if $a_1 ,...,a_R$ are certain matrix valued functions defined on the unit circle, then

$$\det \left( {e^{T\left( {a_1 } \right)} ...e^{T\left( {a_R } \right)} T\left( {e^{ - a_R } ...e^{ - a_1 } } \right)} \right) = \det \left( {T\left( {e^{\tilde a_1 } ...e^{\tilde a_{_R } } e^{T\left( {\tilde a_R } \right)} ...e^{T\left( {\tilde a_1 } \right)} } \right)} \right)$$

$$where\tilde a_r \left( {e^{i{\theta }}} \right) = a_r \left({e^ - i{\theta }} \right)$$