Restrictions of an invertible chaotic operator to its invariant subspaces

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