Factorization of unbounded operators on Köthe spaces

Abstract

Abstract.The main result is that the existence of an unbounded continuous linear operator T between Köthe spaces λ(A) and λ(C) which factors through a third Köthe space λ(B) causes the existence of an unbounded continuous quasidiagonal operator from λ(A) into λ(C) factoring through λ(B) as a product of two continuous quasidiagonal operators. This fact is a factorized analogue of the Dragilev theorem [3,6,7, 2] about the quasidiagonal characterization of the relation (λ(A), λ(B)) ∈ B (which means that all continuous linear operators from λ(A) to λ(B) are bounded). The proof is based on the results of [9] where the bounded factorization property BF is characterized in the spirit of Vogt's [10] characterization of B. As an application, it is shown that the existence of an unbounded factorized operator for a triple of Köthe spaces, under some additonal asumptions, causes the existence of a common basic subspace at least for two of the spaces (this is a factorized analogue of the results for pairs [8,2]).

The existence of a factorized unbounded operator between Fréchet spaces

“…Combining these two results, when the range space has the property (y), common nuclear Köthe subspace is obtained in [3,Proposition 1]. The aim of this note is to prove the Fréchet space analogue of [6,Proposition 6], that is, under the condition that F has property (y), and (E, G, F ) / ∈ BF then there is a common nuclear subspace for all three spaces. We rule out the condition where G can be written as G = ω × X, where X is a Banach space to avoid the case T becomes almost bounded [8].…”

“…Djakov and Ramanujan [1] sharpened this work by removing nuclearity assumption and using a weaker splitting condition. In [6], it is shown that the existence of an unbounded factorized operator for a triple of Köthe spaces, under some assumptions, implies the existence of a common basic subspace for at least two of the spaces. Concerning the class of general Fréchet spaces, the existence of an unbounded operator inbetween is studied in [5].…”