How can i find an estimation of the inverse operator $\|(I - T)^{-1}\|$

Question

$T$ is a bounded linear operator such that $\|T\| \leq e^{a}$; where $a > 0$.\ if the operator $I - T$ is invertible,\ How can i find an estimation of $\|(I - T)^{-1}\|$.

thanks;

Answer 1

The information you have does not give any information on the size of $||(T-I)^{-1}||$:

For every $A<\infty$ there exists $T$ with $||T||<1$, $T-I$ invertible and $||(T-I)^{-1}||>A$.

Proof: Say $0<\epsilon<1/A$. Define $T:\Bbb C\to\Bbb C$ by $$Tz=(1-\epsilon)z.$$