Projections from \(L(E,F)\) onto \(K(E,F)\).

*(English)*Zbl 0912.46011Summary: 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 \(F\).

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 |