Let $A$ be a $C^*$-algebra. If $x \in A$ and $z \in \sigma(x^* x)$, why is $(x^*x - zI)$ neither left nor right invertible?
From Left Invertibility Implies Right Invertibility in Certain $C^*$-algebras, it seems like right invertibility implies left invertibility and vice versa. Unfortunately I don't have a copy of the book in hand, and I failed to prove it by contradiction.
Any help will be appreciated.
Assume $x^*x-zI$ is left invertible, then there is $a\in A$ with $a(x^*x-zI)=I$. But $z$ is a positive real number since $x^*x$ is positive, so taking adjoints yields $(x^*x- zI)a^*=I$ and thus $x^*x-zI$ is right invertible. Hence, it is invertible, contradiction.