Series of operators in \(L^{2}(\mathbb{C})\) that are proportional to unitary ones (Q2258068)

In this work, singular integral operators in \(L^{2}(\mathbb{C})\) of the form \[ Tf=\int_C \frac{(\omega (z)-\omega (\xi))^{n}}{(z-\xi )^{2}} f(\xi) \,d m_{2}(\xi) \] are considered, where \(\mathbb{C}\) is the complex plane, \(m_{2}(\cdot )\) is a Lebesgue measure in \(C\) and \(\omega(\cdot) \) is a function which satisfies certain conditions. It is proved that such an operator is proportional to a unitary one iff \(\omega(z)=az\) or \(\omega(z)=b\bar{z}\).