Does every densely-defined unbounded operator $P$ have a ``factorization'' of the form $$P = A B^{-1},$$ with $A$ and $B$ bounded?
Given two bounded operators $A,B$ such that $B^{-1}$ is a densely-defined unbounded operator, then clearly the composition $P:=AB^{-1}$ is a densely-defined (possibly) unbounded operator (may end up being bounded depending on the relationship between $A$ and $B$). My question is the converse of that.
An answer in either the Hilbert or Banach space setting would be appreciated. Also, additional assumptions on $P$ (like being closed) would be reasonable.
This question is related to factorization of ``transfer functions'' from control theory.