On the invertibility of elementary operators

Let $\mathcal{X}$ be a complex Banach space and $\mathcal{L}(\mathcal{X})$ be the algebra of all bounded linear operators on $\mathcal{X}$. For a given elementary operator $\mathrm{\Phi }$ of length $2$ on $\mathcal{L}(\mathcal{X})$, we determine necessary and sufficient conditions for the existence of a solution of the equation $\mathrm{X}\mathrm{\Phi }=0$ in the algebra of all elementary operators on $\mathcal{L}(\mathcal{X})$. Our approach allows us to characterize some invertible elementary operators of length $2$ whose inverses are elementary operators.