Hypercyclic subspaces of a Banach space

Recently a lot of research has been done on hypercyclicity of a bounded linear operator on a Banach space, based on the hypercyclicity criterion obtained by Kitai in 1982, and independently by Gethner and Shapiro in 1987. By combining this criterion with one extra condition, Montes-Rodríguez obtained in 1996 a sufficient condition for the operator to have a closed infinite dimensional hypercyclic subspace, with a very technical proof. Since then, this result has been used extensively to generate new results on hypercyclic subspaces. In the present paper, we give a simple proof of the result of Montes-Rodríguez, by first establishing a few elementary results about the algebra of operators on a Banach space.