Invertible completions of $2\times 2$ upper triangular operator matrices

Abstract:

In this note we prove that if \begin{equation*}M_{C}=\left (\begin {smallmatrix}A&C\ 0&B\end{smallmatrix} \right ) \end{equation*} is a $2\times 2$ upper triangular operator matrix acting on the Banach space $X\oplus Y$, then $M_{C}$ is invertible for some $C\in \mathcal {L}(Y,X)$ if and only if $A\in \mathcal {L}(X)$ and $B\in \mathcal {L}(Y)$ satisfy the following conditions:

[(i)] $A$ is left invertible;

[(ii)] $B$ is right invertible;

[(iii)] $X/A(X)\cong B^{-1}(0)$.

Furthermore we show that $\sigma (A)\cup \sigma (B)=\sigma (M_{C})\cup W$, where $W$ is the union of certain of the holes in $\sigma (M_{C})$ which happen to be subsets of $\sigma (A)\cap \sigma (B)$.