Let $E,F$ be Banach spaces. Let $\mathcal L(E, F)$ be the space of bounded linear operators from $E$ to $F$, and $\mathcal K(E, F)$ its subspace consisting of compact operators. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. We have Theorem 6.6 in Brezis' Functional Analysis, i.e.,
Fredholm alternative Let $I:E \to E$ be the identity map and $T \in \mathcal K(E, E)$. Then
(a) $\dim N(I-T) < \infty$,
(b) $R(I-T)$ is closed, and more precisely $R(I-T) = N(I-T^*)^\perp$,
(c) $N(I-T) = \{0\} \iff R(I-T) = E$,
(d) $\dim N(I-T)=\dim N(I-T^*)$.
With exactly the same proof as the one of Fredholm alternative, I have showed that
S1 Let $E_0$ be a closed subspace of $E$. Let $I:E_0 \to E$ be the identity map and $T \in \mathcal K(E_0, E)$. Then
(a) $\dim N(I-T) < \infty$,
(b) $R(I-T)$ is closed.
To be honest, I'm not sure whether my proof of S1 is correct or not. There are possibly subtle mistakes that I could not recognize.
S2 Let $E_0$ be a closed subspace of $E$. Let $I:E_0 \to E$ be the identity map and $T \in \mathcal K(E_0, E)$. Then
(a) $\dim N(I-T) < \infty$,
(b) $R(I-T)$ is closed, and more precisely $R(I-T) = N((I-T)^*)^\perp$.
(c) $N(I-T) = \{0\} \iff R(I-T) = E$,
(d) $\dim N(I-T)=\dim N((I-T)^*)$.
My questions are as follows:
Among conclusions (a), (b), (c), (d) of S2, which ones remain true? Can we prove those valid ones by Fredholm alternative?
Thank you so much for your elaboration!
