On the invertibility of elementary operators

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

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