For some reason, I have to work with Trotter product formula recently, but I do not have a strong background in functional analysis.
The following is the statement of the formula from MathWorld
When A and B are self-adjoint operators, $$ e^{t(A+B)} = \lim_{n \to +\infty}(e^{tA/n}e^{tB/n})^n $$
My questions are:
What does the exponential of an operator mean precisely?
How to interpret the convergence? In terms of some norm?
In this case $e^{tA}$ denotes the strongly continuous semigroup generated by the operator $A$ which does not have to be bounded (otherwise it is quite boring).
I suggest you take a look at the book "A Short Course In Operator Semigroups" by Engel and Nagel.
The book is in the yellow sale and available on springerlink.
This formula can be deduced from the Chernoff product formula so you could also look for that. Check out Chapter IV.2.
In the Seventh Internet Seminar 2003/04 semigroups were also treated and the book I'm recommending is based on those notes. Those notes can probably be found online.