Spectrum of T is singleton and and T is normal operator

Question

The following question was asked in my assignment of spectral theory and I am not able to make much progress on it.

Question: Let H be a complex Hilbert Space and let $T\in L(H)$ be normal. If $\sigma(T)$ is singleton {$\lambda$} , then show that $T=\lambda I$.

I am sorry but I am not able to make any progress on how to proceed towards proving this statement. I have been following my class notes.

Can you please give a couple of hints?

Thanks!

Answer 1

By the spectral mapping theorem, if $T$ is a normal operator and $f:\sigma(T)\to\mathbb C$ a continuous function, then $\sigma(f(T))=f(\sigma(T))$. Letting $f(x) = x-\lambda$, we see that $\sigma(T-\lambda I)=0$, and $T-\lambda I$ is normal as well.

Recall that the spectral radius of a normal operator is equal to its operator norm, which immediately implies that $T-\lambda I=0$, or $T=\lambda I$.