Jordan form of extensions of linear operators (Carlson's problem) and minimal reducing subspaces containing a given subspace
type: Item
instance of: scholarly article
title: Jordan form of extensions of linear operators (Carlson's problem) and minimal reducing subspaces containing a given subspace
zbMATH Open document ID: 0939.15005
DOI: 10.1016/S0024-3795(98)10173-8
author: Bernard Charles
author: Abdelkhalek Faouzi
published in: Linear Algebra and its Applications
publication date: 2000-07-02T00:00:00Z
review text: Let \(M\) be the block upper triangular matrix (with complex entries) \(\left( \begin{smallmatrix} A &X\\ 0 &B\end{smallmatrix} \right)\) where \(A\) and \(B\) are given square matrices. The ``Carlson problem'', initiated by \textit{D. Carlson} [Linear Algebra Appl. 5, 293-298 (1972; Zbl 0238.13013)] is to find all the possible Jordan forms of the matrix \(M\). The authors show that the Carlson problem, which is equivalent to the study of Jordan's form of extensions of linear operators in finite dimension, can be reformulated in terms of the description of minimal reducing subspaces containing a given invariant subspace. Using this approach, they study the Carlson problem for the extensions \(M\) (see above) of a nilpotent linear operator \(A\) by a nilpotent cyclic linear operator \(B\) and show that this study is restricted to a certain family of ``irreducible'' extensions for which they give the explicit Jordan form. As a consequence they find a new proof of a result of \textit{C. R. Johnson} and \textit{E. A. Schreiner} [Linear Algebra Appl. 150, 297-314 (1991; Zbl 0725.15009)]. The contents of this paper are part of the doctoral dissertation of the second author.
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 1331333
zbMATH Keywords: minimal reducing subspace
zbMATH Keywords: nilpotent linear operator
zbMATH Keywords: cyclic linear operator
zbMATH Keywords: extensions of linear operators
zbMATH Keywords: block triangular matrices
zbMATH Keywords: Jordan canonical form
zbMATH Keywords: Carlson problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1331333