Homogeneous weakly hyperbolic equations with time dependent analytic coefficients

This paper concerns the Cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients. We give a sufficient condition for the C∞-well-posedness which is also necessary if the space dimension is equal to one.The main point of the paper consists in expressing our condition only in terms of the coefficients of the operator, without needing to know the behavior of the characteristic roots. This is made possible by using the so-called standard symmetrizer of a companion hyperbolic matrix.