Bounded Factorization Property for l-Köthe spaces

Abstract

Let l denote a Banach sequence space with a monotone norm in which the canonical system (e_n)_n is an unconditional basis. We show that the existence of an unbounded continuous linear operator T between l-Köthe spaces lambda^l (????) and lambda^l (????) which factors through a third l-Köthe space lambda^l (????) causes the existence of an unbounded continuous quasidiagonal operator from lambda^l (????) into lambda^l (????) factoring through lambda^l (????) as a product of two continuous quasidiagonal operators. Using this result, we study when the triple (lambda^l (????) , lambda^l (????) , lambda^l (????)) satisfies the bounded factorization property BF (which means that all continuous linear operators from lambda^l (????) into lambda^l (????) factoring through lambda^l (????) are bounded). As another application, we observe that the existence of an unbounded factorized operator for a triple of l-Köthe spaces, under some additional assumptions, causes the existence of a common basic subspace at least for two of the spaces.