Transposition Method for Backward Stochastic Evolution Equations Revisited, and Its Application

The main purpose of this paper is to improve our transposition method to solve both vector-valued and operator-valued backward stochastic evolution equations with a general filtration. As its application, we obtain a general Pontryagin-type maximum principle for optimal controls of stochastic evolution equations in infinite dimensions. In 1articular, we drop the technical assumption appeared in [Q. L"u and X. Zhang, Springer Briefs in Mathematics,Springer, New York, 2014, Theorem 9.1].