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Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know whether it is unitary, self adjoint and normal operator.

How do I proceed?

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The norm is at least $|2+3i|$ by definition, so it is not unitary which implies norm $1$. It can't be selfadjoint, because the eigenvalues of selfadjoint operators must be real. It could be normal though, because $A^*$ being the scalar times the identity is suitable.

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Note, the definition of a normal operator is precisely the property your operator has, namely: $$\|Nx\|=\|N^*\|\text{ for all }x\in\mathcal{D}(N)=\mathcal{D}(N^*)$$ Moreover, this is equivalent to the usually stated: $$N^*N=NN^*\iff N\text{ normal}$$ Try to adjust the proof from that sketch: Normality

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