pr.probability - On operator ranges in Hilbert & Banach spaces

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:

(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 $f(Kg) = mathbb Efg$.

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