Does symmetry of the operator of a dynamical system help stability?

In this article, we address the question of relating the stability properties of an operator with the stability properties of its associate symmetric operator. The linear-algebra results of Bendixson and Hirsch indicate that the symmetric part of a matrix is always less stable than the matrix itself. We show that in a variety of cases, including infinite dimen- sional cases associated to systems of PDEs, the same result is valid. We also discuss the applicability to non-autonomous systems, and we show that, in general, this result is not valid. We also review some of the literature that in these years has appeared on the subject.