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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...

Let ℓ be a Banach sequence space with a monotone norm $\parallel ·{\parallel }_{\ell }$, in which the canonical system $\left(e_i\right)$ is a normalized symmetric basis. We give a complete isomorphic classification of Cartesian products $E_0^{\ell }\left(a\right)\times E_{\infty }^{\ell }\left(b\right)$ where $E_0^{\ell }\left(a\right)=K^{\ell }\left(exp(-p^{-1}{a}_i)\right)$ and $E_{\infty }^{\ell }\left(b\right)=K^{\ell }\left(exp\left(p{a}_i\right)\right)$ are finite and infinite ℓ-power series spaces, respectively. This classification is the generalization of the results by Chalov et al. [Studia Math. 137 (1999)] and Djakov et al. [Michigan Math. J. 43 (1996)] by using the method of compound linear topological invariants developed by the third author....