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Let $H$ be a Hilbert space and $B(H)$ be all the bounded linear operators on $H$, for arbitrary $T\in B(H)$, if $\{P_{i}\}$ is an increasing net of finite-rank projection, can we conclude $P_{i}TP_{i} \rightarrow T$ in norm topology or , even, in strong operator topology?

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    $\begingroup$ You would need $P_i \to 1$ pointwise at the very least $\endgroup$ – Prahlad Vaidyanathan Mar 18 '14 at 3:59
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If $P_i\to I$ in the strong operator topology, then $$ \|(P_iTP_i-T)h\|=\|P_iTP_ih-Th\|\leq\|P_iTP_i-P_iTh\|+\|P_iTh-Th\|\\ \leq\|P_i\|\,\|T\|\,\|(P_i-I)h\|+\|(P_i-I)Th\|\\ \leq\|T\|\,\|(P_i-I)h\|+\|(P_i-I)Th\|\to0 $$ In the norm topology it is never true for infinite-dimensional $H$: just take $T=I$, and then $$ \|P_iTP_i-T\|=\|P_i-I\|=1. $$

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