In this excerpt, it states that ‘ Let $T$ be a bounded linear operator on a Banach space $X$. Suppose spectrum that the $\sigma(T)$ does not separate zero from infinity ( consequently the operator is invertible).’
I wanted to know what is the meaning of the term ‘does not separate zero from infinity’ and how does it imply that $T$ is invertible? An answer or a reference that uses this terminology will be really appreciated.
In the Riemann sphere minus the spectrum, the points $0$ and $\infty$ are in the same connected component. In particular, $0$ is not in the spectrum, so (by definition of spectrum) $T$ is invertible.