Products of commuting nilpotent operators

Abstract

Abstract. We characterize the matrices that are products of two (or more) commuting square-zero matrices and matrices that are products of two commuting nilpotent matrices. We also give a characterization of operators on an infinite dimensional Hilbert space that are products of two (or more) commuting square-zero operators and operators on an infinite-dimensional vector space that are products of two commuting nilpotent operators.1. Introduction. Is every complex singular square matrix a product of two nilpotent matrices? Laffey [5] and Sourour [8] proved that the answer is positive: any complex singular square matrix A (which is not 2 × 2 nilpotent with rank 1) is a product of two nilpotent matrices with ranks both equal to the rank of A. Earlier Wu [9] studied the problem. (Note that [9, Lem. 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].Similar results were proved for the set B(H) of all bounded (linear) operators on an infinite-dimensional separable Hilbert space H. Fong and Sourour [3] proved that every compact operator is a product of two quasinilpotent operators and that a normal operator is a product of two quasinilpotent operators if and only if 0 is in its essential spectrum. 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. Namely, if A = BC and B, C are commuting nilpotent operators then A is nilpotent as well and it commutes with both B and C. If in addition B and C are square-zero then so is A.In the paper we characterize the following sets of matrices and operators:• Matrices that are products of k commuting square-zero matrices for each k ≥ 2.• Matrices that are products of two commuting nilpotent matrices.• Operators on a Hilbert space that are products of k commuting square-zero operators for each k ≥ 2.

A note on Products of Nilpotent Matrices

“…The proof by Laffey relies on preceding results by Wu [7]. It was mentioned before [1] that the right factor on p.229, the last factorization [7, Lemma 3], is in fact not nilpotent for certain values of k. Explicitly, the given factorization of Dg[J k (0), J 2 (0)] is invalid for odd k, since the matrix 0 J 2 (0) J k (0) 0 is not nilpotent when k = 7.…”

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

“…In Tables 1 and 2 Baksalary characterized all situations in which a linear combination of two idempotent matrices is idempotent, which corresponds to cases (1) and (5) in Table 1. [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].…”

Sums and Products of Square-Zero Matrices

“…In [2] it has been shown that an n × n matrix M over an algebraically closed field can be written as a product of two commuting square-zero matrices if and only if M 2 = 0 and rank M ≤ n/4. Also, in [3] it is proven that for any two n × n commuting square-zero matrices A and B over an algebraically closed field, we have rank A + B ≤ n/2.…”