  Pontryagin type maximum principle and Bellman's dynamic programming principle
serve as two of the most important tools in solving optimal control problems.
There is a huge literature on the study of relationship between them. The main
purpose of this paper is to investigate the relationships between Pontryagin
type maximum principle and dynamic programming principle for control systems
governed by stochastic evolution equations in infinite dimensional space, with
the control variables appearing into both the drift and the diffusion terms. To
do so, we first establish dynamic programming principle for those systems
without employing the martingale solutions. Then we establish the desired
relationships in both cases that value function associated is smooth and
nonsmooth. For the nonsmooth case, in particular, by employing the relaxed
transposition solution, we discover the connection between the
superdifferentials and subdifferentials of value function and the first-order
and second-order adjoint equations.
