Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.
Bracic, Janko and Boudi, Nadia.
"On the invertibility of length two elementary operators",
Electronic Journal of Linear Algebra,
Volume 30, pp. 916-913.