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Let $A_1, A_2, ..., A_n, ...$ be linear operators acting on the space $\ell^2$ and defined by:

For $x = (x_1, x_2, ..., x_n, ...) \in \ell^2, A_j(x) = y = (y_1^{(j)}, y_2^{(j)}, ..., y_n^{(j)}, ...),$ where

$y_k^{(j)} = \left\{ \begin{array}{ll} x_k & \quad k \leq j \\ x_k + x_{k-j} & \quad k > j \end{array} \right.$

Given that $A_j$ are bounded, how can I show that the sequence $\{A_j\}$ is not Cauchy in $\mathcal{L}(\ell^2,\ell^2)$?

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Suppose to the contrary that $(A_j)$ is Cauchy. Then $(A_jx)$ is Cauchy for every fixed $x$. But for $x=e^{(1)} := (1,0,0,\ldots)$ it looks like $(A_jx) = e^{(1)}+e^{(j+1)}$ which is definitely not Cauchy.

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