Aplicações comutantes sobre subconjuntos de matrizes.

Abstract

The theory of Functional Identities (FI), introduced in Matej Brešar’s Ph.D. thesis, is rela tively new. Since then, this theory has been developed through a series of papers in which he studied some basic FIs, particularly those concerning the so-called commuting maps. This work is divided into four chapters, where we study additive maps G: Mn(K) → Mn(K) that satisfy the commuting property over some subset A of Mn(K), i.e., G(x)x = xG(x) for all x ∈ A. Such mappings will be called “commuting mappings over A”. Our master’s thesis is based on results from França in [8, 9] and from Xu and Zhu in [23]. Firstly, we present a description of commu ting maps over invertible or singular matrices. As a generalization, we obtain the description of m-additive maps G: Mn(K)m → Mn(K) whose trace T(x) = G(x,...,x) is commuting over the same sets mentioned earlier. Finally, we exhibit an interesting result that states that if T(x) = 0 for all invertible matrices x, then T(Mn(K)) = 0 if one of the following holds: (1) char K = 0; (2) char K > m; (3) char K = m and |K| ̸ = m; (4) |K| g 2m. As a consequence of this last result, We provide an alternative proof for commuting traces of multiadditives on the subset of invertible matrices.