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We study the Cauchy problem for effectively hyperbolic operators $P$ withprincipal symbol $p(t, x,\tau,\xi)$ having triple characteristics on $t = 0$.Under a condition (E) we show that such operators are strongly hyperbolic, thatis the Cauchy problem is well posed for $p(t, x,D_t, D_x) + Q(t, x, D_t, D_x)$with arbitrary lower order term $Q$. The proof is based on energy estimateswith weight $t^{-N}$ for a first order pseudo-differential system, where $N$depends on lower order terms. For our analysis we construct a non-negativedefinite symmetrizer $S(t)$ and we prove a version of Fefferman-Phong typeinequality for ${\rm Re}\, (S(t)U, U)_{L^2({\mathbb R}^n)}$ with a lower bound$-C t^{-1}\|\langle D \rangle^{-1}U\|_{L^2(\mathbb R^n)}$.

Cauchy problem for effectively hyperbolic operators with triple characteristics