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

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