A linear map ( not necessarily bounded ) between normed linear spaces is called a closed operator if its graph is closed. Suppose $X$ is a n.l.s and $Y, Z$ are Banach spaces. Let $A : X_0 ⊆ X → Y$ be a closed operator and $B ∈ B(Z, X)$ such that $Ran(B) ⊆ X_0$ . Prove that $AB ∈ B(Z, Y )$
Since, $Z,Y$ both are given to be Banach, I was willing to apply Closed graph Theorem to $AB$ . Let $(x,y)$ be a limit point of the Graph, then Enough to show that $AB(x)=y$ . Consider any sequence $\{(x_n,ABx_n\}_{n \ge 1}$ in the Graph such that it converges to $(x,y)$ i.e. $x_n \to x$ and $ABx_n \to y$ . Since , B is given to be continuous, $x_n \to x \implies B(x_n) \to B(x)= z $ (say) .
So all we need to show is $Az=y$ . Here I am stuck. How to use the fact that $A$ admits a closed graph?
Thanks in advance for help!
Best Answer
Note that $(Bx_n, ABx_n)$ is a sequence in the graph of $A$. You know $Bx_n \to Bx$ and $ABx_n \to y$ so, since the graph of A is closed, $Bx \in D(A)$ and $ABx =y$. This implies that $x \in D(AB)$ and $ABx =y$ which is what we needed to show that $AB$ has closed graph.