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

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)