Isometric properties of the subspace of compact operators in a space of continuous linear operators
The interesting problem of abstract functional analysis is the problem of Banach spaces E and F for which the space of all compact operators K(E,F) is a nontrivial M-ideal in the space L(E,F) of all bounded (continuous) linear operators. In the paper of Cho and Johnson it has been proved that \(K(E)=K(E,E)\) is an M-ideal in \(L(E)=L(E,E)\) if E is a closed subspace of \(\ell_ p\), \(1<p<\infty\), which satisfies a compact (metrical) approximation property. Some unsolved questions in the mentioned paper of Cho and Johnson are solved in the present paper. A new theorem is also proved in which the condition for a decomposition of the space L(E,F) into a direct sum of the annulator of K(E,F) and a subspace, which is isometrically isomorphic with K(E,F) is given. The consequence of this result is the effective sufficient condition for K(E,F) being the M-ideal in L(E,F). As a corollary the solution of Cho and Johnson's problems is obtained.
scientific article; zbMATH DE number 4136822
scientific article