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Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of $T_{n}$ corresponding to the interval $[0, 1+\delta]$.

  1. Can we verify $||T_{n}-1_{H}||\geq\delta P_{n}^{\perp}$?

  2. Can we get $P_{n}\rightarrow 1_{H}$ in the strong operator topology?

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  1. It is not clear what you mean by that inequality. Assuming you mean $\|T_n-1_H\|\,1_H$: Let $T_n=(1-1/n)\,1_H$. Then $P_n=1_H$ for all $n$. You have $\|T_n-1_H\|=1/n$, and so the inequality you want is $$ \frac1n\,1_H\geq\delta P_N^\perp, $$ which will fail for big enough $n$.

  2. I think you can, but I would have to check my facts carefully. In any case, note that because all the operators involved are projections, WOT convergence implies SOT convergence.

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