When $L\left(E^{*},F^{**}\right)$= $L\left(E^{*},F\right)$+ the adjoints ?

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.

-
I suppose the reason I cannot see that question 23221 has the consequences you state is that I am about ready to fall asleep, so I won't ask about that. Instead, I'll ask this: What is this thing you call $dens$ $H$? –  Harald Hanche-Olsen May 3 '10 at 2:13
Well, does F being reflexive count? –  Matthew Daws May 3 '10 at 9:17
@Harald Hanche-Olsen It's the density character. –  Ady May 3 '10 at 20:50