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

Abstract

LetEEandFFbe 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 spaceL(E,F)L(E,F)of continuous linear operators onto the spaceK(E,F)K(E,F)of compact linear operators. b)L(E,F)=K(E,F)L(E,F)=K(E,F). The answer is positive in certain cases, in particular ifEEorFFhas an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose thatEEandFFare reflexive and thatEEorFFhas the approximation property. Then, ifL(E,F)≠K(E,F)L(E,F)\ne K(E,F), there is no projection of norm 1, fromL(E,F)L(E,F)ontoK(E,F)K(E,F). In this paper, one obtains, in particular, the following result:Theorem.LetFFbe a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such thatF∗F^*has the approximation property. Letλ\lambdabe a real scalar with1>λ>21>\lambda >2. ThenFFcan be equivalently renormed such that, for any projectionPPfromL(F)L(F)ontoK(F)K(F), one has‖P‖≥λ\|P\|\ge \lambda. One gives also various results with two spacesEEandFF.