Factorization of unbounded operators on Köthe spaces

T. Terzioğlu ¹   ; M. Yurdakul ²   ; V. Zahariuta ³

¹ T. Terzioğlu Sabancı University 81474 Tuzla–Istanbul, Turkey
² M. Yurdakul Department of Mathematics Middle East Technical University 06531 Ankara, Turkey
³ V. P. Zahariuta Department of Mathematics Middle East Technical University & Faculty of Engineering and Natural Sciences Sabancı University 81474 Tuzla–Istanbul, Turkey

Studia Mathematica, Tome 161 (2004) no. 1, pp. 61-70

Résumé

The main result is that the existence of an unbounded continuous linear operator $T$ between Köthe spaces $\lambda (A)$ and $\lambda (C)$ which factors through a third Köthe space $\lambda (B)$ 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 $(\lambda (A),\lambda (B)) \in {\mathcal B}$ (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 ${\mathcal BF}$ is characterized in the spirit of Vogt's [10] characterization of ${\mathcal B}$. 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]).

Détail

DOI : 10.4064/sm161-1-4

Keywords: main result existence unbounded continuous linear operator between spaces lambda lambda which factors through third space lambda causes existence unbounded continuous quasidiagonal operator lambda lambda factoring through lambda product continuous quasidiagonal operators factorized analogue dragilev theorem about quasidiagonal characterization relation lambda lambda mathcal which means continuous linear operators lambda lambda bounded proof based results where bounded factorization property mathcal characterized spirit vogts characterization mathcal application shown existence unbounded factorized operator triple spaces under additonal asumptions causes existence common basic subspace least spaces factorized analogue results pairs