  We show that for a general Markov generator the associated square-field (or
carr\'e du champs) operator and all their iterations are positive. The proof is
based on an interpolation between the operators involving the generator and
their semigroups, and an interplay between positivity and convexity on Banach
lattices. Positivity of the square-field operators allows to define a hierarchy
of quadratic and positive energy functionals which decay to zero along
solutions of the corresponding evolution equation. Assuming that the Markov
generator satisfies an operator-theoretic normality condition, the sequence of
energies is log-convex. In particular, this implies polynomial decay in time
for the energy functionals along solutions.
