ON IDEALS OF COMPACT-OPERATORS SATISFYING THE M(R,S)-INEQUALITY

Let r, s is an element of]0, 1]. We prove that a Banach space X satisf ies the M(r, s)-inequality (i.e., \\x**\\greater than or equal to r\\ pi x**\\ + s\\x*** - pi x***\\ For All x*** is an element of X***, wh ere pi is the canonical projection of X** onto X*) whenever r + s/2 > 1 and there exists a norm one projection P on L(X) with Ker P = K(X) perpendicular to satisfying \\f\\ greater than or equal to r\\Pf\\ + s \\f - Pf\\ For All f is an element of L(X). We characterize the last property of K(X) in L(X) by a strong version of the metric compact app roximation property of X. Our results extend some well-known results O n M-ideals. (C) 1998 Academic Press.