Abstract

We consider the Cauchy operator $C$ and the operator of logarithmic potential type $L$ on $L^2(D,d\mu)$, defined by $$ Cf(z)=-\dfrac1\pi\int_D\dfrac{f(\xi)}{\xi-z}\,d\mu(\xi),\quad Lf(z)=- \dfrac1{2\pi}\int_D\log|z-\xi|\,f(\xi)\,d\mu(\xi), $$ where $D$ is the unit disc in $C$, $d\mu(\xi)=h(|\xi|)\,dA$, $h\in L^{\infty}(0,1)$ 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 $I=[0,1]$, $d\nu(r)=rh(r)\,dr$.

Keywords: Cauchy integral operator, operator of logarithmic potential type, space with radial weight.

MSC: 47G10, 45C05

Pages:  135$-$139     

Volume  50 ,  Issue  3$-$4 ,  1998