Operator satisfying certain properties

Question

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?

Answer 1

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.