Jordan form of extensions of linear operators (Carlson's problem) and minimal reducing subspaces containing a given subspace
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.
scientific article; zbMATH DE number 1331333
scientific article