# The spectral projection of a positive operator

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?

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 $n$.