$\| Id+T \| = 1+ \| T \|$ where $\| T \| \in \sigma(T)$

Question

Let $E$ be a banach space and $T$ a continuous linear operator on $E$ with $\| T \| \in \sigma(T)$. Then

$\| Id + T \| = 1 + \| T \|$.

From the triangle inequality, we obtain

$\| Id + T \| \leq \|Id \| + \| T \| = 1 + \|T \|$.

But how do we get the other inequality?

Thanks in advance!

Answer 1

Hint: Note that the spectral radius of $T$ is less than $\|T\|$.