Invertible elements in Banach algebra.

Question

Suppose $A$ is a Banach algebra. Is it true that

a) $x$ and $xy$ are invertible, then so is $yx$;

b) $xy$ and $yx $ are invertible, then so are $x$ and $y$?

Answer 1

I will give you a (very good) hint for the first part.

If $xy$ is invertible, then there is some $z$ such that $(xy)z = 1$. Since $x$ is invertible, we have $x^{-1}xyz = x^{-1}$ but this is nothing more than $yz = x^{-1}$. $z$ is invertible as well so $y = x^{-1}z^{-1}$. I will let you finish it from here - there are only two lines left.

Note that you are interested in $yx$.