Operator norm inequality $\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}$

Question

Let $X, Y$ and $Y^{-1}$ be linear operators on a normed space. How to prove the inequality $$\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}?$$

I already know that $\|XY\|\leq\|X\|\|Y\|$ but I don't see how I can use this fact right now.

Answer 1

Simply note that $$\|XY\|\cdot\|Y^{-1}\|\geq \|XYY^{-1}\|=\|X\|,$$ via the operator norm inequality.

Answer 2

Just notice that $$\Vert X \Vert = \Vert XYY^{-1} \Vert \leq \Vert XY \Vert \Vert Y^{-1} \Vert.$$