The form of a normal operator with only one element in its spectrum Let be $H$ a Hilbert space. Show that if $T$ is a normal linear operator continuous (i.e. $T^*T = TT^*$, with $T^*$ the Hilbert adjunct of $T$) and your spectrum $\sigma(T) = \\{\lambda\\}$, than $T = \lambda I$, when $I : H \to H$ is the identity.

Since $T - \lambda I$ is also normal, we have $$ \| T - \lambda I \| = \text{spr} (T - \lambda I) = 0, $$ showing that $T = \lambda I$.

(I recently asked basically the same question (Self-adjoint operator with single point spectrum), but your formulation is more general so I thought it might be worth sharing the answer here.)