Sums of commuting square-zero transformations
type: Item
instance of: scholarly article
title: Sums of commuting square-zero transformations
zbMATH Open document ID: 1178.15002
author: Nika Novak
published in: Linear Algebra and its Applications
publication date: 2009-10-09T00:00:00Z
review text: This paper is concerned with the problems of characterizing linear transformations that can be expressed as a sum of several special linear transformations. Such problems were first studied by \textit{C. Pearcy} and \textit{D. Topping} [Mich. Math. J. 14, 453--465 (1967; Zbl 0156.38102)] who proved that every bounded linear operator on an infinite-dimensional Hilbert space is a sum of five idempotents as well as of five square-zero operators. \textit{P. Y. Wu} [J. Zemánek (ed.), Functional analysis and operator theory. Proceedings of the 39th semester at the Stefan Banach International Mathematical Center in Warsaw, Poland, held March 2-May 30, 1992. Warsaw: Polish Academy of Sciences, Banach Cent. Publ. 30, 337--361 (1994; Zbl 0814.47001)] showed that a square matrix is a sum of two nilpotent matrices if and only if its trace equals zero. \textit{J.-H. Wang} and \textit{P. Y. Wu} [Stud. Math. 99, No.~2, 115--127 (1991; Zbl 0745.47006)] proved that a matrix \(T\) is a sum of two square-zero matrices if and only if \(T\) is similar to \(-T\). They also showed that a normal (bounded) operator \(T\) on an infinite-dimensional Hilbert space is a sum of two square-zero operators if and only if \(T\) is unitarily equivalent to \(-T\). In this paper, the author shows that a linear transformation on a vector space is a sum of two commuting square-zero transformations if and only if it is a nilpotent transformation with index of nilpotency at most 3 and the codimension of \(\text{im\,}T\cap\ker T\) in \(\ker T\) is greater than or equal to the dimension of the space \(\text{im\,}T^2\). Recall that a linear transformation \(A\) is called a unipotent with index \(n\) if \((A-I)^n=0\) and \((A-I)^{n-1}\not =0\). The author also proves that a linear transformation \(T\) is a product of two commuting unipotent transformations with index 2 if and only if \(T-I\) can be written as a sum of two commuting square-zero transformations.
reviewed by: Wu Jing
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 47A99
zbMATH DE Number: 5612553
zbMATH Keywords: commuting matrices and linear transformations
zbMATH Keywords: square-zero transformations
zbMATH Keywords: square-zero matrices
MaRDI profile type: Publication
full work available at URL: j.laa.2009.06.010
OpenAlex ID: W2162807268
cites work: The Group Generated by Unipotent Operators
cites work: Q3497890
cites work: Products of square-zero operators
cites work: On the closure of the sum of closed subspaces
cites work: Products of nilpotent linear transformations
cites work: Products of unipotent matrices of index 2
cites work: Sums of square-zero operators
cites work: Q4308114
DOI: 10.1016/J.LAA.2009.06.010
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description: scientific article; zbMATH DE number 5612553