Scattering theory and Banach space valued singular integrals (Q2929656)

Let \(H_{0}\) and \(H_{1}\) be two selfadjoint operators in a Hilbert space \(\mathcal{H}\), \(H_{1}=H_{0}+G^{\star}JG\), \(J=J^{\star}\) and \(G\) bounded operators in \(\mathcal{H}\). The authors prove that the wave operators \(W_{\pm}(H_{1},H_{0})\) exist and are complete if (1) for some \(p<\infty\) and some non-negative \(\sigma\)-finite measure \(\nu_{0}\) on \(\mathbb{R}\), one has \(\| GE_{H_{0}}(\delta)G^{\star} \| _{p} \leq \nu_{0}(\delta)\) for all intervals \(\delta\) in \(\mathbb{R}\) (\(E_{H_{0}}\) is the spectral measure associated to \(H_{0}\)) or (2) for some non-negative \(\sigma\)-finite measures \(\nu_{0}\) and \(\nu_{1}\), one has \(\| GE_{H_{0}}(\delta)G^{\star} \| \leq \nu_{0}\) and \(\| GE_{H_{1}}(\delta)G^{\star} \| \leq \nu_{1}\) for all intervals \(\delta\) in \(\mathbb{R}\). The first result generalizes the Kato-Rosenblum theorem and the second one the Kato smoothness theorem. The proofs are based on estimates for the Cauchy transform of operator valued measures.