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 $\mathcal{L}(\ell^2,\ell^2)$?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.