Sums of commuting square-zero transformations
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.
commuting matrices and linear transformations
scientific article; zbMATH DE number 5612553