On Ideals of Compact Operators Satisfying the M r, s.-Inequality

Let r, sgx0, 1x. We prove that a Banach space X satisfies the M r, s.-inequality i.e., 5x***5Gr5p x***5qs5x***yp x***5 ;x***gX***, where p is the canonical projection of X*** onto X*. whenever rqsr2)1 and there exists a norm one projection P on L X.* with Ker PsK X.H satisfying 5f5Gr5Pf5qs5fyPf5 ;fgL X.*. We characterize the last property of K X. in L X. by a strong version of the metric compact approximation property of X. Our results extend some well-known results on M-ideals.