Restrictions of an invertible chaotic operator to its invariant subspaces

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