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Consider $T:C([0,2 \pi]) \rightarrow C([0,2 \pi]) $ $$ T(f(x)) =e^{ix} f(x). $$ Find dual operator $T^*: M([0,2 \pi]) \rightarrow M([0,2 \pi]) $.

Using the Riesz representation for $\mu \in M([0,2\pi])$ and definition of dual operator we can write $$ T^*\mu(f(x)) = \mu(T(f(x)) = \mu(e^{ix} f(x)) = \int_0^{2\pi} e^{ix} f(x) d \mu(x). $$ How can I express $T^* (\mu)$?

Thanks.

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You can use the change of variables formula for measures (see for instance this older StackExchange post: Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure) to rewrite your last expression into the form $$\int_0^{2\pi}f(x) d\nu(x)$$ Obtaining something of the form $$\langle T^* \mu, f\rangle$$ should then allow you to define the operator $T^*$.

JHoogendijk
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