On the spectral properties of the Hilbert transform operator on multi-intervals

Abstract

Let $J,E\subset \mathbb{R}$ be two multi-intervals with non-intersecting interiors. Consider the operator

and let $A^{†}$ be its adjoint. We introduce a self-adjoint operator $\mathcal{K}$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^{†}$. In this paper we study the spectral properties of $\mathcal{K}$ and the operators $A^{†}A$ and $A{A}^{†}$. Our main tool is to obtain the resolvent of $\mathcal{K}$, which is denoted by $\mathcal{R}$, using an appropriate Riemann–Hilbert problem, and then compute the jump and poles of $\mathcal{R}$ in the spectral parameter $\lambda$. We show that the spectrum of $\mathcal{K}$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\mathcal{K}$ consists only of eigenvalues and $0$. If there are common endpoints, then $\mathcal{K}$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\mathcal{K}$ does not have a singular continuous spectrum. The spectral properties of $A^{†}A$ and $A{A}^{†}$, which are very similar to those of $\mathcal{K}$, are obtained as well.