Suppose I have two matrices $A,B$ and let $\rho(\cdot)$ denote the spectral value of the argument. If $\rho(A)<1$ and $\rho(B)<1$, then can I find a matrix norm $\|\cdot\|$ such that $$ \|A\|<1 \qquad\text{and}\qquad \|B\|<1? $$ Thoughts: The spectral radius of a matrix is equal to the infimum over all subordinate norms of a matrix, so for only one matrix we can always find a matrix norm. The question is really about whether we can arbitrarily closely approximate both spectral values with the same norm.
Thank you in advance!
By matrix norm, I assume you mean a norm subordinate to a norm on $\mathbb{R}^n$.
This cannot be true in general because $\rho(AB)\leq ||AB||\leq ||A||.||B||<1$ for a subordinate norm, and the product $AB$ can have arbitrary large spectral radius.
For example, take $$A=\left(\begin{array}{cc} 0 & 2 \\ 0 & 0 \end{array}\right), \; B=\left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right).$$ Their spectral radius are both $0$, but
$$AB=\left(\begin{array}{cc} 2 & 0 \\ 0 & 0 \end{array}\right),$$ whose spectral radius is $2$.