I know
Let $(T_n)$ be a sequence of invertible bounded linear operators on a Hilbert space $H$ converging strongly to a bounded linear operator $T.$ If $\sup_n \|T_n^{-1}\| <\infty$ and $T$ has dense range, then $T$ is invertible and $T_n^{-1}$ converges strongly $T^{-1}.$
I also know that the above result fails if we replace strong convergence by weak convergence. However, I want to prove that if in addition, each $T_n$ is self-adjoint, then the result is true even in weak operator topology. Explicitly, I want to prove:
Let $(T_n)$ be a sequence of self-adjoint bounded linear operators on a Hilbert space $H$ converging weakly to a bounded linear operator $T.$ If $T_n \geq c$ and $T$ has dense range, then $T$ is invertible and $T_n^{-1}$ converges weakly $T^{-1}.$
Following is my attempt:
Clearly, $T$ is also self-adjoint. For $x \in H$ we have, $$\langle x,x\rangle=\langle T_n^{-1}T_nx,x\rangle\leq \frac{1}{c} \langle T_nx,x\rangle\to\frac{1}{c}\langle Tx,x\rangle.$$ This shows that $T$ is injective and has closed range. Since $T$ also has dense range, we may conclude that $T$ is invertible.
Further $$\langle (T_n^{-1}-T^{-1})x,y\rangle=\langle T_n^{-1}(T-T_n)T^{-1}x,y\rangle\leq \frac{1}{c}\langle (T-T_n)T^{-1}x,y\rangle\to0$$ for all $x,y\in H$ and hence $T_n^{-1}$ converges weakly to $T^{-1}.$
Is my solution correct? If not, is the problem statement correct or are there more conditions required to prove the convergence? I will be happy to receive any references with regard to this.