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I need an example of a normal transformation $T$ in an inner product space such that all its eigenvalues are real numbers, but $T\neq T^*$.

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closed as off-topic by Davide Giraudo, Daniel W. Farlow, Alex Provost, TheGeekGreek, user223391 May 14 '17 at 23:40

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I will assume by normal you mean $TT^*=T^*T$. Then you have $T$ is diagonalizable and there exists a unitary operator $P$ such that $PP^*=P^*P=I$ and $T=P^*\Delta P$ where $\Delta$ is a real diagonal matrix and hence $\Delta^ *=\Delta$. Thus we have $T^*=T$. Your example doesn't exist.

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  • $\begingroup$ This assumes finite dimensionality. For the infinite dimensional setting, consider multiplication by $e^{ix}$ in $L^{2}(\mathbb{R})$, which does not have any eigenvalues. $\endgroup$ – Jonas Dahlbæk May 13 '17 at 7:45

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