Expansion of the inverse of positive‐definite Toeplitz operators over polytopes

Let $\tilde\varLambda $ be a polytope in $\mathbb{R}^d$ , $\varLambda =\tilde\varLambda \cap\mathbb{Z}^{d}$ be its trace on the group $\mathbb{Z}^{d}$ and let $T_{\varLambda }(f)$ be a Toeplitz operator, with a positive symbol $f\in L^{\infty}(\mathbb{T}^{d})$ , defined on the Hardy space $H^2(\varLambda )$ . With some additional assumptions on $f$ , a purely algebraic inverse formula of $T_{\varLambda }(f)$ is given. The geometric properties of the operators are highlighted when $\varLambda $ is inflated to get $\varLambda _\lambda =\lambda \varLambda $ with $\lambda=2^{m}$ chosen sufficiently large. A localization result for the inverse operator is obtained. When we are sufficiently close to a $(d-k)$ ‐dimensional face of the inflated polytope, the localized inversion formula is reduced to those operators reflecting this proximity. Subsequently an analogue of the strong Szegő limit theorem is stated; a $d+1$ order asymptotics formula of the trace of the inverse is given that connects each operator of the sum with some geometric measures of our polytope (volume, area, $\,\ldots$ ). Furthermore, the results of Thorsen and Doktorski are retrieved with the help of the Ehrart lattice enumerator formula.