Why is the spectrum of the identity operator on a normed space X is equal to its point spectrum.

Question

Let $I$ be the identity operator on a normed space $X$. I know that $\lambda=1$ is an eigenvalue of $I$. But why is $\lambda=1$ is the only spectral value of $I$? What about the residual spectrum?

Answer 1

If $\lambda\ne1$, then $I-\lambda I=(1-\lambda) I$ is invertible, with inverse $S=\frac1{1-\lambda}\,I$. So any $\lambda\ne1$ is in the resolvent, and $\sigma(I)=\{1\}$.