Solution
= Solution
Hermitian means $T=T^*$, unitary $T^*T=TT^*=I$, and normal $T^*T=TT^*$. If $\|(T-\lambda)x_n\|\to0$ for unit $x_n$, then $\langle Tx_n,x_n\rangle\to\lambda$ and each term is real, so $\lambda\in\mathbb R$. For self-adjoint $T$, a spectral value that is not approximate would make $T-\lambda$ bounded below; its closed range has orthogonal complement $\ker(T-\bar\lambda)=0$, hence is onto, contradicting spectrality.