Relation between spectral radius when the norms are equivalent

Question

If two operator norms are equivalent on B(X), set of all bounded operators on a Banach space X, whether the corresponding spectral radii are the same?

If so, please provide proof or any hint. If not, give some examples. Thanks in advance.

Answer 1

Let $|| \cdot||_1$ and $|| \cdot||_2$ equivalent operator normx on $B(X)$. Hence thereare $a,b >0$ such that

$$a ||A||_1 \le ||A||_2 \le b||A||_1$$

for all $A \in B(X).$ Hence, for $A \in B(X)$, we get

$$a^{1/n} ||A||_1^{1/n} \le ||A||_2^{1/n} \le b^{1/n}||A||_1^{1/n}$$

for all $n \in \mathbb N.$

With $ n \to \infty$ we see that the corresponding spectral radii are the same.