Let $A \in M_n (\mathbb{C})$ and let $\rho (A) = max\{ \mid {\Lambda} : \Lambda$ is an eigenvalue of A \} $

Question

Then is $S = \{ A \in M_n (C) : A = A^*$ and $\rho (A) ≤ 1\}$ is compact in $M_n (\mathbb{R})$?

Answer 1

Hint: by the Spectral Theorem, the spectral radius is equal to the (operator) norm for Hermitian matrices.