Idempotent completions of operator partial matrices
type: Item
instance of: scholarly article
title: Idempotent completions of operator partial matrices
zbMATH Open document ID: 0934.47004
DOI: 10.1007/BF02650728
author: P. Rosenthal
author: Jinchuan Hou
author: H. Radjavi
publication date: 2000-04-10T00:00:00Z
review text: 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 \(AC+ CB= C\).
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A62
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 47B49
zbMATH DE Number: 1356523
zbMATH Keywords: matrix completion problem
zbMATH Keywords: dilation problem
zbMATH Keywords: extension problem
zbMATH Keywords: completions of operator partial matrices
zbMATH Keywords: idempotent extensions
zbMATH Keywords: idempotent dilations
MaRDI profile type: Publication
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1356523