  It is a longstanding unsolved problem to characterize the optimal feedbacks
for general SLQs (i.e., stochastic linear quadratic control problems) with
random coefficients in infinite dimensions; while the same problem but in
finite dimensions was just addressed in a recent work [36]. This paper is
devoted to giving a solution to this problem under some assumptions which can
be verified for several interesting concrete models. More precisely, under
these assumptions, we establish the equivalence between the existence of
optimal feedback operator for infinite dimensional SLQs and the solvability of
the corresponding operator-valued, backward stochastic Riccati equations. A key
contribution of this work is to introduce a suitable notion of solutions (i.e.,
transposition solutions to the aforementioned Riccati equations), which plays a
crucial role in both the statement and the proof of our main result.
