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) = infinity. 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. (C) 2013 Elsevier Inc. All rights reserved.