  We study the forward price dynamics in commodity markets realized as a
process with values in a Hilbert space of absolutely continuous functions
defined by Filipovi\'c. The forward dynamics are defined as the mild solution
of a certain stochastic partial differential equation driven by an infinite
dimensional L\'evy process. It is shown that the associated spot price dynamics
can be expressed as a sum of Ornstein-Uhlenbeck processes, or more generally,
as a sum of certain stationary processes. These results link the possibly
infinite dimensional forward dynamics to classical commodity spot models. We
continue with a detailed analysis of multiplication and integral operators on
the Hilbert spaces and show that Hilbert-Schmidt operators are essentially
integral operators. The covariance operator of the L\'evy process driving the
forward dynamics and the diffusion term can both be specified in terms of such
operators, and we analyse in several examples the consequences on model
dynamics and their probabilistic properties. Also, we represent the forward
price for contracts delivering over a period in terms of an integral operator,
a case being relevant for power and gas markets. In several examples we reduce
our general model to existing commodity spot and forward dynamics.
