Idempotent completions of operator partial matrices (Q1806292)

Continuing the study of the first author on completions of operator partial matrices to (orthogonal) projection [Linear Algebra Appl. 246, 71-82 (1996; Zbl 0864.47018)], the authors of the present paper study the problem of completing the operator partial matrices of the form \(\left[\begin{smallmatrix} A & C\\ ? & ?\end{smallmatrix}\right]\) and \(\left[\begin{smallmatrix} A & C\\ ? & B\end{smallmatrix}\right]\) to idempotents. For example, they show that \(\left[\begin{smallmatrix} A & C\\ ? & ?\end{smallmatrix}\right]\) has an idempotent completion if and only if \(\text{ran}(A- A^2)\subseteq \text{ran }C\) and \(\text{ran }AC\subseteq \text{ran }C\), in which case all such completions can be parametrically represented. They also solve the problem for \(\left[\begin{smallmatrix} A & C\\ ? & ?\end{smallmatrix}\right]\) to have an idempotent completion with norm not greater than a given \(\mu\geq 1\). The previous result leads to the solution of the problem of idempotent extensions and idempotent dilations, namely, the idempotent completion problems for \(\left[\begin{smallmatrix} A & ?\\ 0 & ?\end{smallmatrix}\right]\) and \(\left[\begin{smallmatrix} A & ?\\ ? & ?\end{smallmatrix}\right]\), respectively. Finally, they consider the idempotent completion problem for \(\left[\begin{smallmatrix} A & C\\ ? & B\end{smallmatrix}\right]\): this is solvable if and only if \(\text{ran}(A- A^2)\subseteq \text{ran }C\), \(\text{ran}(B^*- B^{*2})\subseteq \text{ran }C^*\) and \(AC+ CB= C\).