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A bounded operator $T$ on a real or complex Hilbert space is a normal operator when $T^{\ast }T=T{T}^{\ast }$. Equivalently,The forward direction follows by taking inner products. Conversely the norm equality makes $⟨(T^{\ast }T-T{T}^{\ast })x,x⟩=0$ for every $x$, so the polarization identity forces the self-adjoint difference to vanish.

Every Hermitian operator and every unitary operator is normal. The complex diagonal matrix $\mathrm{diag}(2,i)$ is normal but neither self-adjoint nor unitary. In contrast, the unilateral shift $S{e}_n=e_{n+1}$ on the l2 sequence space has $S^{\ast }S=I$ and $S{S}^{\ast }=I-P_{e_1}$, so it is an isometry but not normal. For unbounded operators, normality additionally involves closedness and matching operator domains; the everywhere-defined bounded criterion above cannot be applied while ignoring them.