Products of square-zero operators

Abstract

We characterize matrices that can be written as a product of two or three square-zero matrices. We also consider the same questions for (bounded) operators on an infinite-dimensional, separable, complex Hilbert space and in the Calkin algebra.

On the idempotency, involution and nilpotency of a linear combination of two matrices

“…[2] For the remaining solved problems, the readers are referred to [2,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. For recent advances on this topic and related papers, see [30][31][32][33][34][35][36][37][38][39][40][41][42].…”

Products of commuting nilpotent operators

“…Drnovšek, Müller, and Novak [2] proved that an operator is a product of two quasinilpotent operators if and only if it is not semi-Fredholm. Novak [6] characterized operators that are products of two and of three square-zero operators.Here we consider similar questions for products of commuting square-zero or commuting nilpotent operators on a finite dimensional vector space or on a infinitedimensional Hilbert or vector space. The commutativity condition considerably restricts the set of operators that are such products.…”

“…3] holds but the decomposition given in its proof on [9, p. 229] is not correct since the latter matrix given for the odd case is not always nilpotent.) Novak [6] characterized all singular matrices in M n (F), where F is a field, which are a product of two square-zero matrices. Related problem of existence of k-th root of a nilpotent matrix was studied by Psarrakos in [7].…”

“…Drnovšek, Müller, and Novak [2] proved that an operator is a product of two quasinilpotent operators if and only if it is not semi-Fredholm. Novak [6] characterized operators that are products of two and of three square-zero operators.…”