Matrices de Toeplitz tronqu\'ees sur des polygones convexes. Cas du triangle

TL;DR

This paper investigates the asymptotic behavior of the trace of the inverse of truncated Toeplitz operators on convex polygons, specifically triangles, revealing geometric insights and extending classical determinant evaluation theorems.

Contribution

It introduces a geometric framework for the inverse of Toeplitz operators on triangles and generalizes the Linnik-Szeg"o theorem for these cases.

Findings

Develops a geometric structure for the inverse of Toeplitz operators on triangles.

Provides an asymptotic expansion for the trace of the inverse, highlighting geometric aspects.

Extends the Linnik-Szeg"o theorem to Toeplitz matrices on convex polygons.

Abstract

We consider the class of positive bounded and semi-continuous functions defined on the two dimensional torus If f belongs to this class, then f will be considered as the symbol of a Toeplitz operator truncated on a triangle parametrised by an integer number . We develop a geometric structure of the inverse of the Toeplitz operator and give an asymptotical development of the trace of its inverse wich brings out the geometry of the triangle. The foundation of this result consists in the possibility of f having a factorisation of type |g|^2 where the spectrum of g will be localised in a given semi-cone. This trace theorem allows in particular to find again the Linnik-Szeg\"o theorem about the asymptotical evaluation of the determinant of the truncated Toeplitz operator (or Toeplitz matrix)