Spectral properties of the Cauchy operator and the operator of logarithmic potential type on $L^2$ 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$$Cf(z)=-\dfrac1\piınt_D\dfrac{f(\xi)}{\xi-z}\,d\mu(\xi),\quad Lf(z)=-\dfrac1{2\pi}ınt_D\log|z-\xi|\,f(\xi)\,d\mu(\xi),$$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 $I=[0,1]$, $d\nu(r)=rh(r)\,dr$.