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It is well-known that if $K$ is a convex set of $B(H)$, then the weak operator closure of $K$ equals to the strong operator closure of $K$. [see e.g. Conway Chapter IX, Cor 5.2, A course in Functional analysis]

However, let $U$ be a shift operator on $H=\ell^2$ and $K=conv\{U^k, k\in \mathbb{N}\}$. Consider the WOT closure and SOT closure of $K$. It is clear that $0\in \bar{K}^{wo}$ as $U^k \rightarrow_{wo} 0$. But I don't know how to construct a net of $K$ tending to $0$ in SOT.

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    And what about convexity? – 2017-01-06
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    @TrialAndError conv is the convex hull. Could u give me a specific net tending to 0 in SOT？ – 2017-01-06
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    Aha, i got it. Using convexity is ok by noting that Hilbert is strictly convex. – 2017-01-06
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    There is an optimistic choice for such a net. Take $K_n = \frac{1}{n}\sum_1^n U^k$. You can check that this converges strongly to 0 or you can use a result like the one here: https://math.stackexchange.com/questions/829527/strong-convergence-of-an-averaging-operator?rq=1 – 2017-01-07
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    @JoshKeneda I got it. THX, MAN – 2017-01-08

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