# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Factorization of unbounded operators on Köthe spaces

### Tom 161 / 2004

Studia Mathematica 161(2004), 61-70 MSC: 46A03, 46A13, 46A04, 46A20, 46A32, 47L05. DOI: 10.4064/sm161-1-4

#### Streszczenie

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]).

#### Autorzy

• T. TerzioğluT. Terzioğlu
Sabancı University
81474 Tuzla–Istanbul, Turkey
e-mail
• M. YurdakulM. Yurdakul
Department of Mathematics
Middle East Technical University
06531 Ankara, Turkey
e-mail
• V. ZahariutaV. P. Zahariuta
Department of Mathematics
Middle East Technical University
&
Faculty of Engineering
and Natural Sciences
Sabancı University
81474 Tuzla–Istanbul, Turkey
e-mail

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