Is a right inverse of a element in a unital banach algebra also a left inverse?

Question

I am reading the proof of the following theorem from a set of notes:

I am trying to figure out why the highlighted expression cannot have an inverse. Is it because if it does have an inverse, that would imply $\lambda -x$ has a right inverse which is also its left inverse so it would be invertible which is impossible since $\lambda$ is in the spectrum of $x$. Is that correct?

Answer 1

Note that the two elements in your product commute. ($\lambda$ is a scalar, so it belongs to the center of the algebra) So if the product is invertible then $\lambda-x$ has both a right and left inverse. And it is a basic result that if an element in any unital ring has both a left and right inverse then the element is invertible.