Projections from $L(E,F)$ onto $K(E,F)$

Let $E$ and $F$ be two infinite dimensional real Banach spaces. The following question is classical and long-standing. Are the following properties equivalent?

a) There exists a projection from the space $L(E,F)$ of continuous linear operators onto the space $K(E,F)$ of compact linear operators.

b) $L(E,F)=K(E,F)$.

The answer is positive in certain cases, in particular if $E$ or $F$ has an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose that $E$ and $F$ are reflexive and that $E$ or $F$ has the approximation property. Then, if $L(E,F)\ne K(E,F)$, there is no projection of norm 1, from $L(E,F)$ onto $K(E,F)$. In this paper, one obtains, in particular, the following result:

Theorem. Let $F$ be a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such that $F^*$ has the approximation property. Let $\lambda$ be a real scalar with $1<\lambda<2$. Then $F$ can be equivalently renormed such that, for any projection $P$ from $L(F)$ onto $K(F)$, one has $\|P\|\ge \lambda$. One gives also various results with two spaces $E$ and $F$.