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

Abstract

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)∪I_X) and span(K(Y)∪I_Y), respectively, with r₁+s₁/2>1 and r₂+s₂/2>1. Our results extend some well-known results on M-ideals.