Lemma 1 from [Anderson & Trapp's Shorted Operators, II](http://www.jstor.org/stable/2100462) is
> Let $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:




(1) ran($A$) $\subset$ ran($B$).


(2) $AA^\* \le \lambda^2 BB^\*$ for some $\lambda \ge 0$.


(3) There exists a bounded operator $C$ such that $A = BC$.


Moreover, if (1), (2) and (3) are satisfied, there exists a unique operator $C$ so that ker($A$) = ker($C)$ and ran($C$) $\subset$ closure(ran($B^\*$)).They follow this with the statement, "the lemmas and the original proofs remain valid for operators between two Hilbert spaces." 


**Question:**I would like to know if there is a similar statement for more general Banach spaces, and if so, where I might find it.


**My context:** I am considering the Banach space $\Omega = C(U\_1) \times C(U\_2)$ of continuous functions over two domains. I have a covariance operator $$K : \Omega^\* \to \Omega$$
which is decomposed as $$K = \binom{K\_{11} ~ K\_{12}}{K\_{21} ~ K\_{22}}.$$ I want to apply the above lemma to $A = K\_{21}$ and $B = K\_{22}^{1/2}$. 


**Edit:** If we have a probability measure $\mathbb P$ on $\Omega$, then continuous linear functionals $\Omega^\*$ are random variables. Thus the expectation $\mathbb Efg$ for $f, g \in \Omega^\*$ is well-defined. The covariance operator is the bilinear form defined by $f(Kg) = \mathbb Efg$.



On operator ranges in Hilbert & Banach spaces
(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"](http://www.jstor.org/stable/2038955) 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>


