Okay so let $\displaystyle H$ be a Hilbert Space and let $\displaystyle S,T:H\rightarrow H$ be bounded operators. I need to prove that:

$\displaystyle ST$ is invertible if and only if $\displaystyle S$ and $\displaystyle T$ are both invertible.

I have proved roughly that $\displaystyle S$ and $\displaystyle T$ both invertible implies $\displaystyle ST$ invertible. Not sure how to prove the converse though.

Thanks in advance