This question is somehow related to Is any continuous linear operator from a dual Banach space to a separable Hilbert space the strong-operator limit of a net of adjoint operators of less or equal norm ?. Let $E$ and $F$ be two (say real) Banach spaces, and let

$\mathcal{P}_{E,F}$ =
"*For every linear continuous operator* $T$ *from* $E^{*}$ *to the bidual of* $F$ *there exist some*
$S\in L(E^{*},F)$ *and some* $Q\in L(F^{*},E)$
*such that* $T=Q^{*} +S $ ". Looking at Nonseparable Hilbert spaces as quotients of spaces of bounded functions,
it follows that $\mathcal{P}_{E,F}$

is false if $E=\ell^{1}\left(\Gamma\right)$, and $F=c_{0}\left(\mathbb{N},H\right)$, where $H$ is a Hilbert space, and both $\mid\Gamma\mid$and $dens$ $H$ are big enough. My [maybe rather vague] question would be: are there some known (and non-trivial) sufficient conditions on the pair $\left(E,F\right)$ s.t. $\mathcal{P}_{E,F}$ is true ? Thx in advance.