Spectral perturbation theory and the two weights problem

TL;DR

This paper links spectral perturbation theory to the two weights problem, revealing that Koosis' theorem is a consequence of de Branges' spectral model, thus providing a new operator-valued perspective.

Contribution

It demonstrates that Koosis' theorem can be derived from de Branges' spectral perturbation model, offering a novel operator-theoretic approach to the two weights problem.

Findings

Koosis' theorem is a consequence of de Branges' spectral model.

Spectral perturbation theory provides a new perspective on the two weights problem.

Operator-valued version of Koosis' theorem is established.

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.