Decompositions of linear operators on pre-euclidean spaces by means of graphs

Abstract

In this work we study a linear operator $f$ on a pre-euclidean space $\mathcal{V}$ by using properties of a corresponding graph. Given a basis $\B$ of $\mathcal{V}$, we present a decomposition of $\mathcal{V}$ as an orthogonal direct sum of certain linear subspaces $\{U_i\}_{i \in I}$, each one admitting a basis inherited from $\B$, in such way that $f = \sum_{i \in I}f_i$, being each $f_i$ a linear operator satisfying certain conditions respect with $U_i$. Considering new hypothesis, we assure the existence of an isomorphism between the graphs associated to $f$ relative to two different bases. We also study the minimality of $\mathcal{V}$ by using the graph associated to $f$ relative to $\B$.