  In this paper, we establish some second order necessary/sufficient optimality
conditions for optimal control problems of stochastic evolution equations in
infinite dimensions. The control acts on both the drift and diffusion terms and
the control region is convex. The concepts of relaxed and $V$-transposition
solutions (introduced in our previous works) to operator-valued backward
stochastic evolution equations are employed to derive these optimality
conditions. The correction part of the second order adjoint equation, which
does not appear in the (first order) Pontryagin-type stochastic maximum
principle, plays a fundamental role in our second order optimality conditions.
