The Fredholm alterative says that for $T=I-K$, where $K$ is compact, $T$ is is injective iff $T$ is surjective.
So does the following relation between spectrum $\sigma(T)$ and point spectrum $\sigma_p(T)$ hold? $$\forall\lambda\in\mathbb{C},\ \lambda\neq1:\lambda\in\sigma(T)\iff\lambda\in\sigma_p(T)$$ I know this holds for compact operators $\forall\lambda\in\mathbb{C},\ \lambda\neq0$. Can I really generalize it this way?
Yes, you can generalise it that way. The correspondence between the spectra is not specific to compact operators, if $S$ is any continuous operator and $T = I - S$, then for $\lambda \in \mathbb{C}$ we have
$$\lambda I - T = (\lambda - 1)I + S = -\bigl((1-\lambda)I - S\bigr),$$
so $\lambda \in \sigma(T) \iff 1-\lambda \in \sigma(S)$, $\lambda \in \sigma_P(T) \iff 1-\lambda \in \sigma_P(S)$ and so on, since $-A$ is injective or surjective, has dense image with a bounded inverse etc. if and only if $A$ has that property.