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Let $\mu$ be a positive finite Borel measure on $[-\pi,\pi)$. Define the multiplication operator $M_\mu : L_2(\mu) \to L_2(\mu)$ by $(M_\mu f)(\theta) := e^{i \theta} f(\theta)$. I've proved that $M_\mu$ is an unitary operator; so that each of its eigenvalues can be written as $e^{is}$ for some $s \in [-\pi,\pi)$. Moreover, I've shown that $e^{is}$ is an eigenvalue of $M_\mu$ iff $\mu(\{s\}) > 0$.

My question is regarding the eigenvectors of $M_\mu$ corresponding to the eigenvalue $e^{is}$. I'd like to explicitly describe the set of functions $f \in L_2(\mu)$ satisfying $M_\mu f = e^{is} f$.

I'm a little confused with this description. Any help would be appreciated.

ragrigg
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Let $s$ be an eigenvalue with eigenfunction $f$. Then it holds $$ e^{i\theta} f(\theta) = e^{is} f(\theta) $$ $\mu$-a.e. on $[-\pi,\pi)$. This implies $$ f(\theta) = 0 $$ for all $\theta\ne s$.You can check that any function satisfying this criterion is indeed an eigenfunction, which yields a complete characterization of the eigenfunctions.

daw
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