Structure of the inverse of a Fredholm integral operator of the second kind

Question

NOTE: Cross-posted on Mathematics Stack Exchange

I am trying to solve an equation of the form $$ (\mathbb{I} + K)\phi = f $$ where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\mathbb{R}) $ is a Fredholm integral operator of the second kind, $\mathbb{I}$ is the identity in $L^2([0,1];\mathbb{R})$, and $f$ and $\phi$ are $L^2([0,1];\mathbb{R})$ functions. I am considering, in particular, that the equation I want to solve is $$ \phi(x)+\int_0^1k(x,y)\phi(y) dy = f(x) $$ for a.e. $x\in[0,1]$, with $k\in H^1([0,1]\times[0,1];\mathbb{R})$ (the regularity of the kernel is specific to the problem I am considering but I don't think it hurts).

My question is the following: assuming a solution exists and is unique (i.e. $-1\notin \sigma(K)$), is there any specific structure that we can a priori assume for the inverse operator w.l.o.g. ? For instance, does the solution satisfy an integral equation $$ \phi(x)= f(x) + \int_0^1 l(x,y)f(y) dy $$ for some $l\in H^1([0,1]\times[0,1];\mathbb{R})$ ?

I know that if K were a Volterra integral (with integration limits from $0$ to $x$), one would look for an inverse of the form identity + Volterra integral but I haven't been able to find a similar result spelled out clearly in the case of Fredholm integral equations unless the kernels have some additional structure. I tried to read the original paper by Fredholm "Sur une classe d'équations fonctionnelles" (from Acta Mathematica 27, 1903), yet I am unsure of the sense of convergence for the minors considered in the paper (and thus in what space I should look for my $l$).

Any simpler references to point out (with a more modern notation perhaps) ? I don't want to re-develop what is probably already out there and I want to make sure no counter-examples exist.

Answer 1

Not really an answer, but some remarks:

1. Even if the spectral radius of $K$ is less than $1$, there are counterexamples that the resolvent need not have the required form: This is related to the fact that the composition of two integral operators is not an integral operator, in general. A sufficient condition for the latter is that the operator is regular, that is, that the integral operator with kernel $|k|$ acts in $L^2$ as well. In case $k\in L_2([0,1]^2)$ the integral operator is regular (actually Hilbert-Schmidt). In that case, having spectral radius less than $1$ is certainly sufficient.

2. AFAIK, the general case is still an open problem for regular integral operators. As you observed, the series of Fredholm determinants converges only under additional hypotheses. I think that it is known that it converges in trace norm if the operator is of trace class. I am not sure whether your regularity condition on the kernel is sufficient for the latter.