Decomposing an operator

[shaiguy6]

So I am trying to decompose a linear operator A, in the following manner. I am trying to solve Ax=y for x, and I also have that A=(L-G), so I am trying to solve (L-G)x=y. y is given, and so are L, A, and G. Now, I also know the solutions to Lx1=y and Gx2=0. I'd like to somehow find an expression relating x to x1 and x2. Any help would be appreciated.

[shaiguy6]

In case this wasn't clear, I'll write it over this way:

(L-G)v = y
Lv1=y
Gv2=0

or equivalently:

(L-G)-1y=v
L-1y=v1
G-10=v2

v1 and v2 and v are not equal, they are different. I'd like to have an equation for v in terms of v1 and v2.