## Factorization of unbounded operators on Köthe spaces

### Volume 161 / 2004

#### Abstract

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]).