While studying the action of dilating map of the circle on probability measures, I ran across the following operator:
$$\mathcal{K}^* : L^2_0(S^1)\to L^2_0(S^1)\quad u\mapsto \varphi' u\circ \varphi $$
where $L^2_0(S^1)$ is the Hilbert space of zero mean $L^2$ functions on the circle $S^1=\mathbb{R}/\mathbb{Z}$ and $\varphi$ is a $C^2$ dilating map of $S^1$, of degree $d$ say.

In the model case $\varphi(x)=dx \mod 1$, Fourier analysis shows easily that every $\lambda$ such that $|\lambda| < d$ is an eigenvalue, and that the spectrum is the closed disc of radius $d$.

My question is the following: what can be said in general on the spectrum of $\mathcal{K}^*$? references are welcome if they exist (note that this question led me far away from my usual domain of mathematics, so that this might be a dumb or very easy question -- or might be tough).