  The objective of the paper is to investigate the approximate controllability
property of a linear stochastic control system with values in a separable real
Hilbert space. In a first step we prove the existence and uniqueness for the
solution of the dual linear backward stochastic differential equation. This
equation has the particularity that in addition to an unbounded operator acting
on the Y-component of the solution there is still another one acting on the
Z-component. With the help of this dual equation we then deduce the duality
between approximate controllability and observability. Finally, under the
assumption that the unbounded operator acting on the state process of the
forward equation is an infinitesimal generator of an exponentially stable
semigroup, we show that the generalized Hautus test provides a necessary
condition for the approximate controllability. The paper generalizes former
results by Buckdahn, Quincampoix and Tessitore (2006) and Goreac (2007) from
the finite dimensional to the infinite dimensional case.
