A convergence in norm topology

Question

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then, for any $T_{1}, T_{2}\in B(H)$,

Can we prove that $||P_{n}T_{1}T_{2}P_{n}- P_{n}T_{1}P_{n}T_{2}P_{n}||\rightarrow 0$?

Answer 1

Let $\{E_{kj}\}$ be the canonical matrix units and $P_n=\sum_1^nE_{kk}$. Take $S$ the unilateral shift, $S=\sum_kE_{k+1,k}$, and $T_1=S^*$, $T_2=S$. Then $$ P_nT_1T_2P_n=P_nS^*SP_n=P_n, $$ while $$ P_nS^*P_nSP_n=P_{n-1}. $$ Thus $$ \|P_nT_1T_2P_n-P_nT_1P_nT_2P_n\|=\|P_n-P_{n-1}\|=1 $$ for all $n$.