Given a bounded,
non-negative operator W and a projection P on a Hilbert space, we find necessary
and sufficient conditions for the existence of a non-trivial, non-negative operator V
such that P is bounded from L2(W) to L2(V ). This leads to a vector-valued version
of a theorem of Koosis and Treil’ concerning the boundedness of the Riesz projection
in spaces with weights.