Let $T:L_2[0,1] \to L_2[0,1]$ be a compact linear operator and $S:L_2[0,1] \to L_2[0,1]$ be a bounded linear operator. Prove $$TS-ST \neq e$$ where $e$ is the identity operator.
My solution.
I am aware of the fact that if $T$ is compact and $S$ is bounded then both $TS$ and $ST$ are compact. I am also aware of the fact that the sum of compact operators is compact, hence $TS-ST$ is compact.
However I know that the identity operator is not compact in infinite dimensional spaces, therefore $TS-ST$ cannot be equal to $e$ since one is compact and other is not.
Is my solution correct?