Sums of square-zero infinite matrices

Sums of square-zero endomorphisms of a free module

“…A recent result by Slowik [8] states that every operator on a vector space of countably infinite dimension is a sum of ten square-zero operators. In other words, one has k ≤ 10 in Question 1 if R is a field.…”

“…It is known that every such operator A is a sum of five square-zero bounded operators; also, A can be written as a sum of four square-zero bounded operators if and only if A is a commutator [10]. Slowik [8] and de Seguins Pazzis [6] considered this problem in a different setting of column-finite matrices, that is, for endomorphisms of an infinite-dimensional vector space. This paper is devoted to a further generalization of this problem.…”

“…The problem of decomposing an endomorphism into a sum of square-zero endomorphisms is being studied for fifty years [2,4,5,6,8,10], and there exist several well-known results on this topic. The most widely studied case is that of endomorphisms of a finite-dimensional vector space, which correspond to n × n matrices over a field.…”

Sums of Square-Zero Infinite Matrices Revisited

“…In this short note we will improve the result from [7,8] and prove the following: Theorem 1.1 Let F be a field of characteristic different from 2. Every N × N column finite matrix can be written as a sum of at most 10 square-zero matrices.…”

“…In [7,8] it was shown that every N × N column finite, i.e., having only a finite number of nonzero entries in each column, matrix defined over a field F such that char(F) = 2 can be written as a sum of (at most) 12 square-zero matrices.…”