On the existence of a factorized unbounded operator between Fr\'echet spaces

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 $(y)$.