On operator ranges in Hilbert & Banach spaces

One way to reformulate (1) as a factorization result is this:

Suppose $S:X\to Z$ and $T:Y\to Z$ are bounded linear operators and $SX\subset TX$. Then $S$ factors through the map $T$ induces from $Y/T^{-1}(0)$ into $Z$. To see this, WLOG $T$ and $S$ are one to one, and just observe that by e.g. the closed graph theorem $SX\subset TX$ implies that for some $a$, $SB_X \subset aTB_Y$.

Of course, this implies that if $T^{-1}(0)$ is complemented in $Y$, then $S$ factors through $T$ itself.

(1) does not generally imply (3) for bounded operators between Banach spaces. The first example I have a reference for was due to Douglas and was included in "Factorization of operators on Banach space" by Embry in 1973. That paper has much more that might interest you, such as the fact that a factorization holds when you have the reverse range inclusion of the adjoints.

See also: http://www.jstor.org/stable/2043114