Restrictions of an invertible chaotic operator to its invariant subspaces

Let M be a closed subspace of a separable, infinite dimensional Hilbert space H with dim ( H / M ) = ∞ . We show that a bounded linear operator A : M → M has an invertible chaotic extension T : H → H if and only if A is bounded below. Motivated by our result, we further show that A : M → M has a chaotic Fredholm extension T : H → H if and only if A is left semi-Fredholm.