The conditions of selfadjointness of the operator of singular integration (Q1179073)

Let \(\Gamma\) be a closed rectifiable curve and let \(L_ 2(\Gamma,\rho)\) be the weighted Hilbert space with the inner product \[ \langle\varphi,\psi\rangle=\int_ \Gamma\varphi(t)\overline \psi(t)\rho(t)| dt|. \] Suppose that the contour \(\Gamma\) and the weight \(\rho\) are taken so that the operator of singular integration \[ (S\varphi)(t)={1\over \pi i}\int_ \Gamma{\varphi(\tau)d\tau\over\tau- t} \] for \(t\in \Gamma\) is bounded in \(L_ 2(\Gamma,\rho)\). It is proved that \(S\) is self-adjoint iff \(\Gamma\) if a circle and \(\rho(t)\) = constant.