Spectral properties of the Cauchy operator and the operator of logarithmic potential type on L2 space with radial weight

Abstract

We consider the Cauchy operator $C$ and the operator of logarithmic potential type $L$ on $L^2(D,d\mu )$, defined by$$Cf(z)=-{\displaystyle \frac{1}{\pi }}ın{t}_D{\displaystyle \frac{f(\xi )}{\xi -z}}d\mu (\xi ),Lf(z)=-{\displaystyle \frac{1}{2\pi }}ın{t}_D\log |z-\xi |f(\xi )d\mu (\xi ),$$where $D$ is the unit disc in $C$, $d\mu (\xi )=h(|\xi |)dA$, $hın{L}^{ınfty}(0,1)$ is a function, positive a.e.\ on $(0,1)$ and $dA$ the Lebesgue measure on~$D$. We describe all eigenvectors and eigenvalues of these operators in terms of some operators acting on $L^2(I,d\nu )$ with $I=[0,1]$, $d\nu (r)=rh(r)dr$.