I know this is a fairly general question, but I would like to know anything I can about obtaining the inverse of an exponential of a lie operator. More specifically, I want to know how one can constructively define the following operators:
$$\frac{1}{e^A},~~~\frac{1}{1+e^{A_1}+e^{A_2}+\cdots}, ~~~ \frac{1}{f(e^{A_i})}$$
where $A_i$, $i=1,2,\cdots$ are Lie operators.
EDIT:
@avs pointed out in his original answer what exactly those inverse operators are (i.e. how one can define them) but in a non-constructive way. I am specifically looking for ways of constructing representations of those inverse operators.
If I understand the question correctly, if we have any linear operator $A$ for which the exponential $\exp(A)$ is meaningful, then $A$ commutes with $-A$, and so $$ \exp(A) \exp(-A) = \exp(A + (-A)) = \exp(\mbox{the zero operator}) = I. $$
The inverse of $(I + B)^{-1}$ (again, if the latter is defined) is $(I+B)$.