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Let $L(H)$ be the space of bounded operators on some Hilbert space.

We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT). It is immediate that the norm topology is stronger than the SOT which is again stronger than the WOT.

My question is now the following. Given a linear map $T:L(H) \rightarrow L(H).$ Are there any relations of the kind: If this operator is WOT-WOT continuous, is it automatically SOT-SOT continuous or are similar relations true?

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  • $\begingroup$ @BjørnKjos-Hanssen that is for $C_0-$semigroups $\endgroup$
    – Zwars
    Jan 24, 2017 at 7:33
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    $\begingroup$ WOT-WOT continuity implies norm-norm continuity by the closed graph theorem (for the Banach space $L(H)$ with the operator norm norm). $\endgroup$
    – Jochen Wengenroth
    Jan 24, 2017 at 11:15

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