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This paper concerns the spectral properties of the Neumann-Poincar\'eoperator on $m$-fold rotationally symmetric planar domains. An $m$-foldrotationally symmetric simply connected domain $D$ is realized as the$m$th-root transform of a certain domain, say $\Omega$. We prove that thedomain of definition of the Neumann-Poincar\'e operator on $D$ is decomposedinto invariant subspaces and the spectrum on one of them is the exact copy ofthe spectrum on $\Omega$. It implies in particular that the spectrum on thetransformed domain $D$ contains the spectrum on the original domain $\Omega$counting multiplicities.

Spectral properties of the Neumann-Poincar\'e operator on rotationally symmetric domains in two dimensions