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My question seems to be easy but I cannot spot the answer. I am interested in ranges of operators defined on $c_0$. The celebrated "operator version" of Sobczyk's theorem says that if we are given a separable Banach space $X$ and its subspace $Y$, then every bounded operator $T\colon Y\to c_0$ can be extended to a bounded operator $\overline{T}$ with $\|\overline{T}\|\leq 2\|T\|$ (categorically speaking, $c_0$ is "separably injective"). I am wondering if I could use this theorem (or anything else) to (dis)prove the following conjecture:

If $X$ is a $c_0$-saturated separable Banach, then the range of every operator $T\colon c_0\to X$ embeds into $c_0$. We know that (consult Lindenstrauss and Tzafriri's book) every quotient of $c_0$ embeds into $c_0$ but how about ranges of this sort of operators?

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    There is a quite recent [survey article](http://matematicas.unex.es/~fcabello/files/printable/21.pdf) on Sobczyk's theorem by Cabello Sánchez, Castillo and Yost. While I haven't read it in detail, it might contain some pointers to the literature.2011-06-16
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    I'm really not sure if I understand your question correctly, but as it is stated I don't really see how the information that $X$ contains a complemented copy of $c_0$ should help. If $T: c_0 \to Y$ is *any* operator to a separable Banach space, simply consider $X = Y \oplus c_0$ and compose $T$ with the inclusion $Y \to X$. The range of $T$ will remain the same and certainly $X$ is separable. Also, it would be nice if you could make your question a bit more precise (e.g. what exactly does it mean for the range of $T$ to *embed* into $c_0$)?2011-06-16
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    $T(X)$ isomorphic to a subspace of $c_0$. Right, I was thinking about $c_0$-saturation, since if $X$ contains a copy $c_0$ then in the separable setting it is automatically complemented. Btw, I know this paper quite well.2011-06-16

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