I'm solving the next exercise about adjoints and linear extensions:
Let $T:\mathcal{D}(T)\rightarrow\ell^{2}$ be defined by $$y=(\eta_{j})=Tx,\quad\eta_{j}=j\xi_{j},\quad x=(\xi_{j}) $$ where $\mathcal{D}(T)\subset\ell^{2}$ consists of all $x=(\xi_{j})$ with only finitely many nonzero terms $\xi$. (a) Show that T is unbounded. (b) Does T have proper linear extensions? (c) Can T be linearly extended to the whole space $\ell^{2}$?
To prove a) is enough to consider sequence $\xi_{j}=\frac{1}{j^{2}}$ $\forall j\in\{1,\ldots,n\}$ and $\xi_{j}=0$ $\forall j\geq n+1.$
My problems are in b) and c).
In b), I consider $s=\{(a_{n})_{n\in\mathbb{N}}\subset\mathbb{C}:\displaystyle\lim_{n\rightarrow\infty}n^{2}a_{n}=0\}\subset\ell^{2}$ such that $\mathcal{D}(T)\subset s.$ So we defined $\hat{T}:s\rightarrow \ell^{2}$ by $\hat{T}(x)=(\eta_{j})$ as before. Then $\hat{T}$ is a proper linear extension of $T.$
For c) I'm stuck. I don't get any useful if I suppose that there is such extention.
Any kind of help is thanked in advanced.