Bulletin of the
Korean Mathematical Society BKMS

It is proved that if $2\leq p \leq q < \infty$, $q \neq 2$, and $X$ is a closed subspace of $\sumXp$ ($\dim X_n < \infty$) which has the compact approximation property then $K(X,\Lq)$ is an M-ideal in $L(X,\Lq)$. Suppose that $Y$ is a Banach space such that there exists a sequence $\seq{K_n}$ in $K(Y)$ satisfying $K_n \to I_Y$ strongly and $\norm{I_Y - 2K_n}\to 1$. Then for any Banach space $X$, $K(X,Y)$ is an M-ideal in $L(X,Y)$ if and only if every contractive operator $T:X\to Y$ has property (\Ms) which is a dual notion of property (M) introduced by N.~Kalton and D.~Werner.

Keywords: compact approximation property, compact operator, M-ideal, operator

MSC numbers: 46A32, 41A50