Supremum of product of matrices with norm less than one

Question

We consider 2 invertible matrices $A_{1}$ and $A_{2}$, and a submultiplicative norm $\|.\|$ such that $\|A_{1}\|\le 1$ and $\|A_{2}\|\le 1$. My goal is to find: $$ \sup\limits_{i_{1}\in \mathbb N^{*},i_{2}\in \mathbb N^{*},i_{3}\in \mathbb N}\|A_{1}^{i_1}A_{2}^{i_2}A_{1}^{i_3}\| $$

My intuition says that the supremum is achieved when $i_{1}=1, i_{2}=1,i_{3}=0$. I tried to prove this by induction but no luck. Does any one can help?

Answer 1

Using the submultiplicativity of the norm, if $p,q,r$ are three integers such that $p,q\ge1$ and $r\ge0$, then \begin{aligned} \|A_1^pA_2^qA_1^r\| &\le\|A_1^pA_2^q\|\|A_1\|^r\\ &\le\|A_1^pA_2^q\|\\ &\le\|A_1^pA_2\|\|A_2\|^{q-1}\\ &\le\|A_1^pA_2\|\\ &\le\|A_1\|^{p-1}\|A_1A_2\|\\ &\le\|A_1A_2\|. \end{aligned}