Let $\left( S\_{n}^{1}\right) $ and $\left( S\_{n}^{2}\right) $ two sequences
of operators in $\mathcal{L}(E\_{1},F\_{1})$ and $\mathcal{L}(E\_{2},F\_{2})$
where $E\_{i},F\_{i},i=1,2$ are Hilbert spaces such that $\left\Vert
S\_{n}^{1}x-S^{1}x\right\Vert \_{F\_{1}}\underset{n\rightarrow \infty }{%
\rightarrow }0,$ $\forall x\in E\_{1}$ for some $S^{1}\in \mathcal{L}%
(E\_{1},F\_{1})$ and such that
$$
\ker S\_{n}^{1}=\{0\}\Leftrightarrow \ker S\_{n}^{2}=\{0\},\text{ }\forall
n\geq 0.
$$
Assume that $\ker S^{1}=\{0\}.$ Does this implies that $\ker S^{2}=\{0\}?.$
Thank you in advance.



Kernels of sequences of operators
No, even if $\|S^2\_n x-S^2x\|\_{F\_2}\to0$ for some $S^2\in\mathcal L(E\_2,F\_2)$ and all $x\in E\_2$.


Indeed, let e.g. $E\_1=F\_1=E\_2=F\_2=H:=\ell^2$, $S^1\_n=I$, $S^1=I$, $S^2=0$, and $S^2\_n x=(e^{-|k-2n|}x\_k)\_{k=1}^\infty$ for $x=(x\_k)\_{k=1}^\infty\in\ell^2$. Then all the conditions hold. In particular,
$$\|S^2\_n x\|\_H^2\le e^{-n}\sum\_{k=1}^n|x\_k|^2+\sum\_{k=n+1}^\infty|x\_k|^2\to0$$
as $n\to\infty$.


However, the conclusion $\text{ker}S^2=\{0\}$ does not hold.


