Inequality of matrix norm

Question

Suppose that we have four matrices $Y, A, B, C$. We have $Y = ABC$. The power of $Y$ is $Y^{i} = (ABC)^{i} $.

1. By using the submultiplicative property of matrix norm, $\| Y^{i}\| \le \| A\|^{i} \| B\|^{i} \|C \|^{i} $ can be easily proved.
2. However, is the relation $\| Y^{i}\| \le \| A^{i}\| \| B^{i}\| \|C^{i} \| $ holds? Can anybody give the proof?

Answer 1

This inequality does not hold: if one of those matrices is nilpotent, then $Y^i$ might not be null, but the rhs of the inequality is zero.

Take, e.g., $A=\pmatrix{0&1\\0&0}$, $B=I_2$, $C=A^T$. Then $Y^i= \pmatrix{1&0\\0&0}$ but $A^i=C^i=0$ for $i\ge 2$.