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Projections from $$L(E,F)$$ onto $$K(E,F)$$. (English) Zbl 0912.46011
Summary: 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 $$L(E,F)\neq K(E,F)$$, 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 $$\| P\| \geq \lambda$$. One gives also various results with two spaces $$E$$ and $$F$$.

##### MSC:
 46B20 Geometry and structure of normed linear spaces 46B28 Spaces of operators; tensor products; approximation properties
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