Lemma 1 from Anderson & Trapp's Shorted Operators, II is

Let $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:They follow this with the statement, "the lemmas and the original proofs remain valid for operators between two Hilbert spaces."(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($B^\*$)).

**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 $f(Kg) = \mathbb Efg$.