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

### Marco Bertola

Concordia University and Université de Montréal, Canada; SISSA/ISAS, Trieste, Italy### Alexander Katsevich

University of Central Florida, Orlando, USA### Alexander Tovbis

University of Central Florida, Orlando, USA

## Abstract

Let $J,E⊂R$ be two multi-intervals with non-intersecting interiors. Consider the operator

and let $A_{†}$ be its adjoint. We introduce a self-adjoint operator $K$ acting on $L_{2}(E)⊕L_{2}(J)$, whose off-diagonal blocks consist of $A$ and $A_{†}$. In this paper we study the spectral properties of $K$ and the operators $A_{†}A$ and $AA_{†}$. Our main tool is to obtain the resolvent of $K$, which is denoted by $R$, using an appropriate Riemann–Hilbert problem, and then compute the jump and poles of $R$ in the spectral parameter $λ$. We show that the spectrum of $K$ 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 $K$ consists only of eigenvalues and $0$. If there are common endpoints, then $K$ 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, $K$ does not have a singular continuous spectrum. The spectral properties of $A_{†}A$ and $AA_{†}$, which are very similar to those of $K$, are obtained as well.

## Cite this article

Marco Bertola, Alexander Katsevich, Alexander Tovbis, On the spectral properties of the Hilbert transform operator on multi-intervals. J. Spectr. Theory 12 (2022), no. 2, pp. 339–379

DOI 10.4171/JST/403