# Operator satisfying certain properties

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?

• How about $S_n=0$ fro all $n$? – Kabo Murphy Aug 17 at 4:52
• Just noticed a major typo, we want the strong limit to be NOT compact. @KaviRamaMurthy you are correct in the original case – Evan Gorman Aug 17 at 14:55

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.