  For locally convex spaces $X$ and $Y$, the continuous linear map $T:X \to Y$
is called bounded if there is a zero neighborhood $U$ of $X$ such that $T(U)$
is bounded in $Y$. Our main result is that the existence of an unbounded
operator $T$ between Fr\'echet spaces $E$ and $F$ which factors through a third
Fr\'echet space $G$ ends up with the fact that the triple $(E, G, F)$ has an
infinite dimensional closed common nuclear K\"othe subspace, provided that $F$
has the property $(y)$.
