Adjoint mapping

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I need to show the following: let $\phi$ be a linear function $V \rightarrow V$, $V$ with finite dimensions and with $\phi^*$ as adjoint function, than $\phi$ is normal if and only if there is a polynomial $f$ so that $$f(\phi)=\phi^*$$

can you help me out?

Here is normal as $\phi \circ \phi^*=\phi^*\circ\phi$ to intend. Thank you