This paper is devoted to the study of the Cauchy problem in$C$^{∞}and in the Gevrey classes for some second order degenerate hyperbolic equations with time dependent coefficients and lower order terms satisfying a suitable Levi condition.

We prove that if$T$is a strictly singular one-to-one operator defined on an infinite dimensional Banach space$X$, then for every infinite dimensional subspace$Y$of$X$there exists an infinite dimensional subspace$Z$of$X$such that$Z∩Y$is infinite dimensional,$Z$contains orbits of$T$of every finite length and the restriction of$T$to$Z$is a compact operator.

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