Chongman Cho

Abstract Suppose X and Y are closed subspaces of (Σ X n )p and (ΣY n ) q (1 < p ≦ q < ∞, dim Xn < ∞, dim Y n < ∞), respectively. If K(X, Y) , the space of the compact linear operators from X to Y , is dense in L(X, Y) , the space of the bounded linear operators from X to Y , in the strong operator topology, then K(X, Y) is an M -ideal in L(X, Y) .

Suppose X and Y are Banach spaces for which K(X;Y ), the space of compact operators from X to Y , is an M-ideal in L(X;Y ), the space of bounded linear operators from X to Y. If Z is a closed subspace of Y such that L(X;Z) has property SU in L(X;Y ) and d(T;K(X;Z)) = d(T;K(X;Y )) for all T 2 L(X;Z), then K(X;Z) is an M-ideal in L(X;Z) if and only if it has property SU in L(X;Z).

For 1 < p , r < ∞ , , { n i } bounded, the space K ( X ) of all compact operators on X is the only nontrivial M ‐ideal in the space B ( X ) of all bounded linear operators on X .

Let $X$ be a Banach space and $Z$ a closed subspace of a Banach space $Y.$ Denote by $\CL(X,Y)$ the space of all bounded linear operators from $X$ to $Y$ and by $\CK(X,Y)$ its subspace of compact linear operators. Using Hahn-Banach extension operators corresponding to ideal projections, we prove that if either $X^{**}$ or $Y^*$ has the Radon-Nikod$\acute{y}$m property and $\CK(X,Y)$ is an $M$-ideal (resp. an $HB$-subspace) in $\CL(X,Y),$ then $\CK(X,Z)$ is also an $M$-ideal (resp. $HB$-subspace)