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> **Definition.** Let us define a Banach space $X$ to be *co-Sobczyk* if every linear bounded operator $T:Z\to c\_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c\_0$.
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By the classical Sobczyk Theorem, each separable Banach space is co-Sobczyk.
But the class of co-Sobczyk spaces includes many non-separable Banach spaces. 
In particular, a Banach space $X$ is co-Sobczyk if each separable subspace of $X$ is contained in a complemented separable subspace. So, all classical Banach spaces $c\_0(\Gamma)$ and $\ell\_p(\Gamma)$ for $1\le p<\infty$, are co-Sobczyk for any set $\Gamma$.


I have a strong feeling that co-Sobczyk spaces have been studied in the theory of non-separable Banach spaces, so asking the MO commubnity for a proper reference and an existing terminology (I suspect that co-Sobczyk spaces are called differently). 



What is a name for co-Sobczyk Banach spaces?
Nigel Kalton studied [a similar but stronger notion](https://books.google.cz/books?id=8pwDl-jWe4YC&pg=PA160&lpg=PA160&dq=c0%20EXTENSION%20PROPERTY&source=bl&ots=WUHwUcGc6p&sig=ACfU3U0YLIKq0yi7CpFgXHW4EiG7H_E7cQ&hl=pl&sa=X&ved=2ahUKEwismPy-ytTiAhXh0qYKHa7oCcwQ6AEwCnoECAUQAQ#v=onepage&q=c0%20EXTENSION%20PROPERTY&f=false): for $\lambda \geqslant 1$, he termed a Banach space $X$ to have the $(\lambda, \mathcal{C})$-*extension property*, when for any compact space $K$ you may find extensions of operators $T$ from subspaces of $X$ into $C(K)$ to operators from $X$ to $C(K)$ with norm at most $\lambda \|T\|$.


It is thus natural to term your spaces as having the $(\lambda, c\_0)$-*separable extension property* if you care about the extension constant.


Update: [Correa and Tausk](https://www.sciencedirect.com/science/article/pii/S0022247X13002540) call this *separable $c\_0$-extension property*.


