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Let $H$ be a Hilbert space over $\mathbb{C}$. Let $A:H\rightarrow H$ be a normal operator ($AA^{*}=A^{*}A$). Let $\lambda\in\mathbb{C}$ be such that $Ax=\lambda x$ for some $x\neq 0$ in $H$. Is it true that $A^*x=\bar{\lambda}x$ ?

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Yes, If $A$ is normal, then so is $N=A-\lambda I$, and $$ \|Nx\|^2 = \langle Nx,Nx \rangle = \langle N^*Nx,x\rangle = \langle NN^*x,x\rangle = \langle N^*x,N^*x\rangle = \|N^*x\|^2. $$ Therefore, $Nx=0$ (i.e., $Ax=\lambda x$) iff $N^*=0$ (i.e., $A^*x=\overline{\lambda}x$.)

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