  The present paper continues the study of infinite dimensional calculus via
regularization, started by C. Di Girolami and the second named author,
introducing the notion of "weak Dirichlet process" in this context. Such a
process $\X$, taking values in a Hilbert space $H$, is the sum of a local
martingale and a suitable "orthogonal" process. The new concept is shown to be
useful in several contexts and directions. On one side, the mentioned
decomposition appears to be a substitute of an It\^o type formula applied to
$f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ function
and, on the other side, the idea of weak Dirichlet process fits the widely used
notion of "mild solution" for stochastic PDE. As a specific application, we
provide a verification theorem for stochastic optimal control problems whose
state equation is an infinite dimensional stochastic evolution equation.
