Linear operators on S-graded vector spaces Original Research Article

The aim of this paper is to formulate and study natural generalizations of the well-known classical classification problems of linear algebra. We first consider the problem about one linear operator which acts on a finite-dimensional vector space graded by a partially ordered set with involution S=(A,*). For a fixed S and a fixed polynomial f(t), we study the problem of classifying (up to S-similarity, which is defined in a natural way) the operators phi satisfying f(phi)=0; in particular, a complete description of tame and wild cases is obtained. Furthermore, we prove that there are no new tame cases in the “most” general situation when objects of a Krull–Schmidt subcategory of mod k are considered instead of graded spaces. We consider also a “most” general natural extension of the problem on the reduction of the matrix of a linear map by means of elementary row and column transformations. Finally, we introduce the notion of “dispersing representation of a quiver”; in terms of these representations one can formulate many classification problems and, in particular, all the known and new ones encountered in this paper.