Spectral perturbation theory and the two weights problem Alexander PushnitskiAlexander Volberg 47G1047A40Two weights problemscattering theoryoperator valued weights The famous two weights problem consists of characterising all possible pairs of weights such that the Hardy projection is bounded between the corresponding weighted $L^2$ spaces. Koosis's theorem of 1980 gives a way to construct a certain class of pairs of weights. We show that Koosis's 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's model provides an operator-valued version of Koosis's theorem. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5389 10.1512/iumj.2014.63.5389 en Indiana Univ. Math. J. 63 (2014) 1349 - 1364 state-of-the-art mathematics http://iumj.org/access/