Problem about convergence of operator norm and compactness of an operator.

Question

Problem-$1$ Given the sequence of continuous linear operators $T_n : l^2 \to l^2$ defined by $$T_n(x) = (0, 0, \ldots, x_{n+1}, x_{n+2}, \ldots)$$ for every $x \in l^2$. Then for every $x \neq 0$ in $l^2$ i want to check whether $\|T_n\|$ and $\|T_nx\|$ converge to $0$ or not?

Problem-$2$ Let the continuous operator $T: l^2 \to l^2$ defined by $$T_n(x) = (0, x_1, 0, x_3, \ldots,)$$ for every $x = (x_1, x_2, \ldots) \in l^2$. To find whether $T$ is compact or not?

According to me for $(1)$, $$\|T_nx\|^2 = \sum_{k = n+1}^{\infty} \|x_k\|^2$$ Now as $x \in l^2$, so there exist $N \in \mathbb{N}$ such that $$\sum_{k = N+1}^{\infty} \|x_k\|^2 < \epsilon$$ we conclude that $\|T_n(x)\| \to 0$. What about the others? am i correct?

Answer 1

$||T_n x|| \to 0$ for each $x$, but $||T_n e_{n+1}||=1$ so $||T_n|| \geq 1$ for each $n$. For 2) the answer is no: $Te_1=e_2,Te_3=e_4,...$ so $\{T{e_{n+1}}\}$ has no convergent subsequence. Hence $T$ is not compact.

Answer 2

What you did for problem 1 is correct (maybe it would be better to say that for any positive $\varepsilon$, there exists an $N$ such that...). Your computation shows that $\left\lVert T_n\right\rVert\leqslant 1$. Considering $x$ as the vector whose $(n+1)$th coordinate is $1$ and all the others are zero, you can show the reverse inequality.

For problem 2, the range of the unit ball contains the collections of elementary vectors $e_{2i+1}$, where $e_j$ is the element of $\ell^2$ whose $j$-th coordinate is $1$ and all the others are zero. Is $\left\{e_{2i+1},i\in\mathbb N\right\}$ compact?