Factorization of unbounded operators on Köthe spaces
Terzioğlu, Tosun and Yurdakul, M. and Zakharyuta, Vyacheslav (2004) Factorization of unbounded operators on Köthe spaces. Studia Mathematica, 161 (1). pp. 61-70. ISSN 0039-3223 (Print) 1730-6337 (Online)
Official URL: http://dx.doi.org/10.4064/sm161-1-4
The main result is that the existence of an unbounded continuous linear operator T between Kothe spaces lambda(A) and lambda(C) which factors through a third Kothe space A(B) causes the existence of an unbounded continuous quasidiagonal operator from lambda(A) into lambda(C) factoring through lambda(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 (lambda(A), lambda(B)) is an element of B (which means that all continuous linear operators from lambda(A) to lambda(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  characterization of B. As an application, it is shown that the existence of an unbounded factorized operator for a triple of Kothe 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]).
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