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Spectral properties of the Cauchy operator and the operator of logarithmic potential type on \(L^2\) space with radial weight. (English) Zbl 1029.47505

The author considers the Cauchy integral operator \(C\) and the operator of logarithmic potential type \(L\) on \(L^2(D,d\mu)\), defined by \(Cf(z)=-\pi^{-1}\int_Df(\xi)(\xi-z)^{-1} d\mu(\xi)\) and \(Lf(z)=-(2\pi)^{-1}\int_D\ln|z-\xi|f(\xi) d\mu(\xi)\), where \(D\) is the unit disc in \(\mathbb C\), \(d\mu(\xi)=h(|\xi|) dA\), \(h\in L^{\infty}(0,1)\) is a function, positive a.e. on \((0,1)\) and \(dA\) the Lebesgue measure on \(D\). Spectral properties of these operators in the case \(h\equiv 1\) where studied in [J. M. Anderson, D. Khavinson and V. Lomonosov, Q. J. Math., Oxf. II. Ser. 43, 387-407 (1992; Zbl 0764.31001)] and [J. Arazy and D. Khavinson, Integral Equ. Oper. Theory 15, 901-919 (1992; Zbl 0779.47015)]. Eigenvectors and eigenvalues of operators \(C\) and \(L\) are described in terms of certain operators acting on \(L^2(I,d\nu)\) where \(I=[0,1]\), \(d\nu(r)=rh(r) dr\).

MSC:

47G10 Integral operators
45C05 Eigenvalue problems for integral equations
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