Is $QT=T|_{N(T)^{\perp}}$, for linear operator $T:H_1\to H_2$ and orthog0nal projection $Q: H_2\to \overline{R(T)}$?

Question

Let $T: H_1\to H_2$ be a bounded linear operator, where $H_1$ and $H_2$ are Hilbert spaces. Let $Q$ be the orthogonal projection of $H_2$ onto $\overline{R(T)}$. Is it correct to say that $QT=T^\#$, where $T^\#=T|_{N(T)^{\perp}}$

Answer 1

No. All you are doing by taking $QT$ is changing the codomain, if you intend $Q$ to have codomain $\overline{R(T)}$ (otherwise you are changing nothing). It won't change the domain. If $x$ is nonzero and in $N(T)$, then $QTx=Tx=0$, while $T|_{N(T)^\perp}(x)$ doesn't exist because $x$ is not in the domain.