$A+B=AB$ does it follows that $AB=BA$? If $A$, $B$ are two normal operators such that: $A+B=AB$ does it follow that $AB=BA$?

In the finite dimensional case:
We have $(A-I)$ invertible because
$$(A-I)(B-I)=AB-B-A+I=AB-(A+B)+I=I.$$
Note that this doesn't show invertibility if our vector space is infinite dimensional. To convince yourself with matrices, take the determinant of the above expression.
Now since $(A-I)B=A$,
$$A(A-I)B=A^2=(A-I)BA.$$
Because $A(A-I)=(A-I)A$, this means 
$$(A-I)AB=(A-I)BA.$$
Invertibility was shown so we cancel to obtain the result.
