Diagonalizability of nonautonomous linear systems with bounded continuous coefficients

A linear ODE system with bounded continuous coefficients is studied. Provided that the diagonal terms are “separated” and the off-diagonal terms satisfy a smallness threshold, we present some sufficient conditions for the diagonalizability of the system. The results are valid for any couple of Banach subspaces V , W of B C ( R , C n ) admissible for a family of reduced ( n − 1 dimensional) linear homogeneous systems having exponential dichotomy.