  The famous two weights problem consists in characterising all possible pairs
of weights such that the Hardy projection is bounded between the corresponding
weighted $L^2$ spaces. Koosis' theorem of 1980 gives a way to construct a
certain class of pairs of weights. We show that Koosis' theorem is closely
related to (in fact, is a direct consequence of) a spectral perturbation model
suggested by de Branges in 1962. Further, we show that de Branges' model
provides an operator-valued version of Koosis' theorem.
