norm of matrix powers if spectral radius is at most 1

Question

Let $A$ be a matrix whose spectral radius $\rho(A) < 1$. Can anything be said about $\|A^2\|_2, \|A^3\|_2, \dots$ in relation to $\|A\|_2$?

I know that $\|A^k\| \le \|A\|^k$ but I am looking for something else. Can it be said that $\|A^k\|_2 \le \|A\|_2$?

The other result I could find is from these notes which say that $A^k \rightarrow 0$.

Answer 1

The condition $\rho(A) < 1$ for $n \times n$ matrices does not restrict $\|A^k\|_2$ for any $k < n$. For example, if $A$ has entries $t$ on the first super-diagonal and $0$ everywhere else (i.e. $t$ times a single Jordan block of size $n$ for eigenvalue $0$), we have $\|A^k \|_2 = |t|^k$ for $1 \le k \le n-1$ but $\rho(A) = 0$.