M(r,s)-inequality for K(X,Y) in L(X,Y)

We study Banach spaces X and Y for which the subspace of all compact operators K(X,Y) forms an ideal satisfying the M(r,s)-inequality in the space of all continuous linear operators L(X,Y). We prove that K(X,Y) is an M(r12r2,s12s2)- and an M(r1r22,s1s22)-ideal in L(X,Y) whenever K(X) and L(Y) are M(r1,s1)- and M(r2,s2)-ideals in span(K(X)∪IX) and span(K(Y)∪IY), respectively, with r1+s1/2>1 and r2+s2/2>1. Our results extend some well-known results on M-ideals.