Weighted integral Hankel operators with continuous spectrum

Abstract

Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in $L^2({\mathbb{R}}_{+})$. These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel $s^{\alpha }{t}^{\alpha }(s+t{)}^{-1-2\alpha }$, where $\alpha >-1/2$. Our analysis can be considered as an extension of J.Howland's 1992 paper which dealt with the unweighted case, corresponding to $\alpha =0$.