Geodesic
A series of operators in $L^2(\mathbb C)$ proportional to unitary ones
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 188-201

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We prove that singular integral operators in $L^2(\mathbb C)$ defined by the formula $$ Tf(z)=\int_\mathbb C\frac{(w(z)-w(\xi))^n}{(z-\xi)^{n+2}}f(\xi)\,dm_2(\xi), $$ where $|w(z)-w(\xi)|\leq c|z-\xi|$, $z,\xi\in\mathbb C,$ are proportional to unitary ones if and only if $w(z)=az$ or $w(z)= b\overline z$. 
@article{ZNSL_2013_416_a10,
     author = {N. A. Shirokov},
     title = {A series of operators in $L^2(\mathbb C)$ proportional to unitary ones},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {188--201},
     publisher = {mathdoc},
     volume = {416},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_416_a10/}
}
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N. A. Shirokov. A series of operators in $L^2(\mathbb C)$ proportional to unitary ones. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 188-201. http://geodesic.mathdoc.fr/item/ZNSL_2013_416_a10/