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Give an example of a Hilbert space $H$ and a sequence of compact operators $(S_n)_{n=1}^{\infty}$ on H such that:

(i) $||S_n||\leq 1$

(ii) The operators $V_N=\sum_{n=1}^N\dfrac{1}{n}S_n$ converge strongly as $N\rightarrow\infty$

(iii) The strong limit of the operators $V_N$ is not compact

I've tried working on $H=\ell^2(\mathbb{R})$ defining $S_n$ as various projections but I cannot seem to satisfy all of the conditions at once. Any hints?

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  • $\begingroup$ How about $S_n=0$ fro all $n$? $\endgroup$ – Kabo Murphy Aug 17 at 4:52
  • $\begingroup$ Just noticed a major typo, we want the strong limit to be NOT compact. @KaviRamaMurthy you are correct in the original case $\endgroup$ – Evan Gorman Aug 17 at 14:55
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Let $ {e_n}$ be the usual basis for $l^{2}$ and $S_n$ be the projection on the span of $e_i: i\leq n$. Then $S_n$ converges to the identity operator strongly so this has all the desired properties.

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