Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 106 / 4 / b

If $T$ is normal, then $S=T-I$ is normal and $\sigma (S)=\{0\}$. The C-star identity and normality imply $\Vert S^n\Vert =\Vert S{\Vert }^n$. The spectral radius formula consequently givesso $T=I$.

Normality is essential. On ${\ell }^2$, let $N$ send $e_2$ to $e_1$ and every other basis vector to zero. Then $N\ne 0$ but $N^2=0$, sowhile $I+N\ne I$.