Invertible completions of 2×2 upper triangular operator matrices

In this note we prove that if M C = ( A a m p ; C 0 a m p ; B ) {equation*}MC= ( {smallmatrix}A&C\ 0&B{smallmatrix} ) {equation*} is a 2 × 2 2× 2 upper triangular operator matrix acting on the Banach space X ⊕ Y X⊕ Y , then M C MC is invertible for some C ∈ L ( Y , X ) C∈ L(Y,X) if and only if A ∈ L ( X ) A∈ L(X) and B ∈ L ( Y ) B∈ L(Y) satisfy the following conditions: [(i)] A A is left invertible; [(ii)] B B is right invertible; [(iii)] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X slash upper