Uniform Convergence of uniformly bounded Operators

Question

let $X,Y$ be Hilbertspaces with continuous embedding $X\hookrightarrow Y$ and $T_n$ a family of uniformly bounded linear operators, i.e. \begin{align*} \exists c>0\;:\|T_n\|_{\mathcal{L}(X,Y)} \leq c \quad \forall n\in\mathbb{N}. \end{align*} Moreover, the operators should converge pointwise, i.e. \begin{align*} T_nx \to x \text{ in } Y \quad \forall x\in X \end{align*} Now, let $\{x_m\}_{m}\subset X$ be a bounded sequence. Does \begin{align*} \max_{m\in\mathbb{N}} \|x_m - T_nx_m\|_Y \to 0 \text{ as } n\to\infty \end{align*} hold? Thank you all.

Answer 1

No. Basically, specialize to the case $X=Y=\ell^2$ (separable), call $S_n=id-T_n$ and consider a dense sequence $\{x^m\}_{m\in\Bbb N}\in B(0,1)$. You are then asking if a bounded sequence $S_n$ such that $S_nx\to 0$ for all $x$ must satisfy $\lVert S_n\rVert\to 0$. This is not the case: call $S$ the backwards shift $(Sx)_k=x_{k+1}$ and consider $S_n=S^n$. $S^nx\to 0$ for all $x$ but $\lVert S^n\rVert=1$.