# Generalized Fredholm eigenvalue problem?

The standard homogeneous Fredholm integral equation of the second-kind is:

$$K[f]=\lambda f$$

where $K$ is an integral operator:

$$K[f](x) = \int_0^1 K(x,y)f(y)\mathrm d y$$

In analogy with the generalized eigenvalue problem of finite matrices, I can consider an equation of the form:

$$K[f] = \lambda A[f]$$

where $A$ is a linear operator. Has this problem been considered in the literature? Is there a name for it?

Yes, the Fredholm alternative for integral equations is a special case of a more general theory.

See for instance Kato's Perturbation theory for linear operators, Chapter IV, Theorem 5.26:

If $T$ is semi-Fredholm (so $\dim \ker T < \infty$ or $\operatorname{codim}\operatorname{ra} \, T < \infty$) and $A$ is $T$-compact (so $(x_n)$ and $(Tx_n)$ bounded implies that $(Ax_n)$ has a convergent subsequence), then $S = T+A$ is also semi-Fredholm with $\operatorname{ind} S = \operatorname{ind} T$.

The classical Fredholm alternative is a special case of this theorem: Choosing $T$ as the identity, $T$ is clearly (semi-) Fredholm with $\operatorname{ind} T = 0$, and the operator $A$ is $T$-compact for $T$ if and only if it is compact in the usual sense. The theorem then shows that for the identity $T$ and compact $A$, the operator $T+A$ still has index $0$, which is the content of the Fredholm alternative (it is injective if and only if it is surjective).