I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
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. Let $T \in \mathcal L(E, F)$. We denote by $R(T)$ and $N(T)$ the range and kernel of $T$.
- Prove that the following properties are equivalent:
- (A) $\dim N(T) < \infty$ and $R(T)$ is closed.
- (B) There exist a constant $C>0$ and a finite-rank operator $P \in \mathcal L(E, E)$ such that $P=P^2$ and $|u|_E \le C (|Tu|_F + |Pu|_E)$ for all $u \in E$.
- (C) There exist a Banach space $G$, an operator $Q \in \mathcal K(E, G)$, and a constant $C>0$ such that $|u|_E \le C (|Tu|_F + |Qu|_G)$ for all $u \in E$.
- Assume that $T$ satisfies (A). Prove that $(T+S)$ also satisfies (A) for every $S \in \mathcal K(E, F)$.
There are possibly subtle mistakes that I could not recognize in below attempt of (2). Could you please have a check on it? Is there a direct approach that does not appeal to (1)?
By (1), there exist a Banach space $G$, an operator $Q \in \mathcal K(E, G)$, and a constant $C>0$ such that $|u|_E \le C (|Tu|_F + |Qu|_G)$ for all $u \in E$. Let $G' := F \times G$. We endow $G'$ with $1$-product norm, i.e., $|(u, v)|_{G'} = |u|_{F} + |v|_G$. Then $G'$ is a Banach space. We consider a bounded linear operator $$ Q':E \to G', u \mapsto (Su, Qu). $$
The topology induced by $|\cdot|_{G'}$ coincides with the product topology of those induced by $|\cdot|_{F}$ and $|\cdot|_{G}$. Because $S,Q$ are compact operators, so is $Q'$. For $u \in E$, $$ \begin{align} |u|_E &\le C (|Tu|_F + |Qu|_G) \\ &\le C (|(T+S)u|_F + |Su|_F + |Qu|_G) \\ &= C (|(T+S)u|_F + |Q'u|_{G'}). \end{align} $$
Then $(T+S)$ satisfies (C). The claim then follows from (1).
