Spectral perturbation theory and the two weights problem (1306.1912v1)

Abstract: 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.