We can factor $g$ into a product of irreducible elements.
Note that the apparently [= seemingly] infinite product in the denominator is in fact finite.
Then $P$ is the product of several integer factors of about $x^n$ in size.
We need to check that $F$-derivatives behave in the way we expect with regard to sums, scalar multiples and products.
A different notation is used because the usual tensor product symbol is reserved for the tensor product of $A$-bimodules.
Then $B(E)$ is a unital Banach algebra with product the composition of operators.