If I have factors of linear operators say
$$(a_1 + A)(a_2 + A)(a_3 + A)\cdots(a_n + A) = 0$$
$A$ being an linear operator(i guess it really doesn't matter its operator or not) why
$$\sum_{n} \frac{X_n(A)}{X_n(a_n)} = 1$$
$X$ function is the quotient function, for example:
$$X_2(A) = (a_1 + A)(a_3 + A)\cdots(a_n + A)$$
a numerical example is $$(1+A)(1-A)=0$$
then
$$\frac{X_1(A)}{X_1(a_1)} +\frac{X_2(A)}{X_2(a_2)} = \frac{1+A}{2} +\frac{1-A}{2} = 1$$