Simultaneous chaotic extensions for general operators on a Hilbert subspace

For an infinite-codimensional closed subspace M of a separable Hilbert space H, we show that every bounded linear operator A:M→H has a chaotic extension T:H→H. As a generalization of this result, we further show that for any uniformly bounded sequence of linear operators An:M→H, there exists a bounded linear operator V:M⊥→H on the orthogonal complement M⊥ of M that simultaneously extends all operators An to chaotic operators An+V:H→H.