Factorisation de fonctions positives sur le tore: Applications à l’inverse des opérateurs de Toeplitz tronqués

We consider the class of positive bounded and lower semi-continuous functions defined on the torus T 2 . A factorisation theorem provides an equality f = g g ¯ where g has its spectrum in a given cone. Hence the notion of Λ -factorisation where Λ is a convex polygon. From such a factorisation we derive an inversion formula for the truncated Toeplitz operator T Λ ( f ) . These results are applied to obtain a trace theorem for T Λ ( f ) - 1 if Λ is a triangle. The trace is related to geometric properties of the triangle.