Let $X$ be a separable Hilbert space and $P_n,P:X\to X$ continuous projections (you can assume they are orthogonal projections if it helps) such that $P_n\to P$ in the operator norm and the range of $(I-P_n)$ and $(I-P)$ has finite rank. Assume that $A:X\to X$ is a bounded injective operator. Note that the sets $A^{-1}P_n(X)$ and $A^{-1}P(X)$ are closed in $X$ and therefore there exists orthogonal projections $Q_n$ and $Q$ onto theses subspaces. It is true that $Q_n\to Q$ in the operator norm?
Thank you so much for your help!
The conclusion is not true.
Let $\mathcal{H}=\ell^2(\mathbb{N}_0)$ and $\{e_k\}_{k=0}^\infty$ denote the standard orthonormal basis. Consider the shift operator $$A(x_0,x_1,\ldots )=(0,x_0,x_1,\ldots )$$ Let $P_n$ be the orthogonal projection onto the orthogonal complement of the element $v_n=\cos(n^{-1})e_0-\sin(n^{-1})e_1.$ We have $$P_ny=y-\langle y,v_n \rangle v_n$$ As $v_n\to e_0$ we obtain $P_n\to P,$ where $Py=y-\langle y,e_0\rangle e_0.$ Observe that $$P_n\mathcal{H}= \{y\in \ell^2\,:\, y_1=\cot(n^{-1})y_0\}$$ Therefore $$A^{-1}(P_n\mathcal{H}) =\{x\in \ell^2\,:\, x_0=0\}$$ On the other hand $$A^{-1}(P\mathcal{H})=\ell^2(\mathbb{N}_0)$$ Hence $Q_n\neq I$ while $Q=I.$