Interpretation of Fredholm Alternative with respect to PDEs

Question

I have studied the Fredholm Alternative, which states the following:

Theorem: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator on $H$. Then:

1.$N(I-K)$ is finite dimensional.

2.$R(I-K)$ is closed.

3.$R(I-K) = N(I-K^{*})^{\perp}$

4.$N(I-K) = \{0\}$ <=> $R(I-K) = H$

How does the theorem imply the following with respect to partial differential equations:

$(\alpha) = \{\text{for each }f\in H, \text{ the equation $u - Ku = f$ has a unique solution.}\}$

or else

$(\beta) = \{\text{The homogenous equation } u - Ku = 0 \text{ has solutions }u \neq 0 \}$

Thanks

Answer 1

$\alpha$ implies that $I - K$ is bijective, hence injective, giving in turn "not $\beta$".

Not $\beta$ says that $I - K$ is injective, hence you get the bijectivity from 4. This gives $\alpha$.