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Let $A,B$ be two closed operators on a Hilbert space. Let $P_i$ be an increasing net of projections with $P_i\uparrow {\bf 1}$. If $(A-B)P_i =0$ for every $i$ (if necessary, we may assume that $AP_i,BP_i$ are bounded and self-adjoint for every $i$), do we have $A-B=0$?

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  • $\begingroup$ I don't quite get your question; do you meant $A - B = 0$ or perhaps $A = B$? I mean, what about $A - B$? Cheers! $\endgroup$ – Robert Lewis Mar 30 '18 at 6:05
  • $\begingroup$ Ah, sorry. I meant A-B=0 @RobertLewis $\endgroup$ – user92646 Mar 30 '18 at 6:11
  • $\begingroup$ Would you mind to give the definition of closed operator? Is it continuous? $\endgroup$ – C.Ding Apr 8 '18 at 10:16
  • $\begingroup$ they are just general closed operators. Sorry, I can not give a specific definition @C.Ding $\endgroup$ – user92646 Apr 9 '18 at 4:53
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Multiplication is a separately continuous map in WOT, SOT and norm.

I would guess the same is true for any reasonable locally convex topology in B(H).

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  • $\begingroup$ what topology do we have for closed operators? $\endgroup$ – user92646 Mar 30 '18 at 13:45
  • $\begingroup$ I completely overlook that your operators are closed instead of bounded. $\endgroup$ – Nick Bottom Mar 30 '18 at 15:11

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