Inverse of an unbounded operator

Question

I have 3 unbounded operators $R,S$ and $T$ defined sur $L^{2}(\mathbb{R})$ such that:

1)$\forall u\in D(T)=D(R)\cap D(S): ||Tu||2=||Ru||^2+||Su||^2$.

2) $T$ is invertible on $D(T)\to L^{2}(\mathbb{R})$.

3)$RS\not=SR$.

Can we say that $S$ and $R$ are invertibles.

Thank you in adcance

Answer 1

No, not even if $T,R,S$ are all bounded can you say this (let $R$ be the projection onto a subspace and $S$ onto its orthogonal complement).

Indeed if $T$ is invertible (bounded or not) and $p$ is an orthogonal projection you've got with $R:=pT$, $S:=(1-p)T$ that $$\|Tx\|^2=\|Rx\|^2+\|Sx\|^2$$ for all $x\in D(T)$. But $R$ and $S$ have non-zero kernels.