If I am given $Y\in \mathbb{C}^{n \times n}$ is nonsingular and a matrix norm defined as $\|A\|_Y = \|Y^{-1}AY\|_2$, how do I show that it has the property: $$\|AB\|_Y \leq \|A\|_Y\|B\|_Y$$
At first I was thinking that $ \|Y^{-1}AY\|_2$ is equal to the max singular value of $A$ because it looks like a similarity transform; however, I don't think that is valid.
Hint:
Note that\begin{align} \|AB\|_Y &= \|Y^{-1}ABY\|_2\\ &=\|(Y^{-1}AY)(Y^{-1}BY)\|_2 \end{align}
Try to complete it using submultiplicative property.