# For two closed operators $A,B$, can we say that $P_i\uparrow 1$ and $(A-B)P_i =0$ implies $A-B$?

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 $A-B=0$?

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