Suppose we have a trace class kernel $K_x(z,z')$ acting on $L^2((ax,bx))$ ( square integrable functions on the interval $(ax,bx)$, where $x>0$ is a positive parameter).
In trying to study the properties of the Fredholm determinant $$F_1(x)=\det(I-K_x(z,z'))_{L^2(ax,bx)}$$ i have found that it is easier to first rescale it to the fixed interval $(a,b)$, in other words look at the determinant $$F_2(x)= \det(I-K_x(zx,z'x))_{L^2(a,b)} $$
Then one should only have to prove a relation between the two. I initially thought this had to be equality but I am confused as to how to write this down.
The question is how are these determinants related??
Many thanks in advance!