Consequences of the existence of a non-compact operator between nuclear Köthe spaces

Abstract

It is proved that if a continuous, linear, non-compact operator TɛnK(A)→K(B) exists between two nuclear Kothe spaces then there exists a common (up to isomorphism) step space of the two spaces provided K(A) is regular and K(B) is isomorphic to a subspace of K(A) or K(B) is regular and K(A) is isomorphic to a quotient space of K(B) or K(A) and K(B) satisfy a certain splitting condition. Consequences in some particular cases are also obtained.