K(X,Y) as subspace complemented of L(X,Y)

TL;DR

This paper investigates the structure of operator spaces between Banach spaces, showing conditions under which certain subspaces are complemented and exploring properties like the Radon-Nikodym property.

Contribution

It establishes new conditions for complemented subspaces in operator spaces and analyzes the Radon-Nikodym property in these contexts.

Findings

K(X,Y) contains a complemented c0 if Y contains c0 and sequences in Y have w* convergent subsequences

Relations between projections in L(X,Y) and K(X,Y) are characterized when Y has the approximation property

Results on the Radon-Nikodym property in L(X,Y) are obtained

Abstract

Let X,Y be two Banach spaces ; in the first part of this work, we show that K(X,Y) contains a complemented copy of c0 if Y contains a copy of c0 and each bounded sequence in Y has a subsequece which is w* convergente. Afterward we obtain some results of M.Feder and G.Emmanuele: Finally in this part we study the relation between the existence of projection from L(X,Y) on K(X,Y) and the existence of pro- jection from K(X,Y ) on K(X,Y) if Y has the approximation property. In the second part we study the Radon-Nikodym property in L(X,Y):