  Siegel's $E$- and $G$-functions were defined in two conjecturally equivalent
senses, strict and broad. By taking up and completing a sketch of Andr\'e, we
state and prove the analogue in the broad sense of the Andr\'e-Chudnovsky-Katz
theorem, which is a structure theorem on the $G$-operators in the broad sense
(they are differential operators cancelling the $G$-functions in the strict
sense). We deduce from that a structure theorem on the $E$-operators in the
broad sense, which are differential operators cancelling the $E$-functions in
the broad sense. As an application of this last theorem, we give a new proof of
a generalization by Andr\'e of the Siegel-Shidlovskii theorem on the algebraic
independence of the values of $E$-functions in the broad sense.
