  In this paper we discuss some aspects of the Heisenberg uncertainty relation,
mostly from the point of view of non self-adjoint operators. Some equivalence
results, and some refinements of the inequality, are deduced, and some relevant
examples are discussed.
  We also begin a sort of {\em dynamical analysis} of the relation, in
connection with what has been recently called $\gamma$-{dynamics} and
$\gamma$-symmetries, and we discuss in some details the role of different
scalar products in our analysis.
  The case of self-adjoint operators is recovered as a special case of our
general settings.
