Remarks on bounded operators in ℓ-Köthe spaces

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For locally convex spaces X and Y, the continuous linear map T:X → Y is said to be bounded if it maps zero neighborhoods of X into bounded sets of Y. We denote (X,Y) ∈ B when every operator between X and Y is bounded. For a Banach space ℓ with a monotone norm · in which the canonical system (eₙ) forms an unconditional basis, we consider ℓ-Köthe spaces as a generalization of usual Köthe spaces. In this note, we characterize ℓ-Köthe spaces ℓ(apn) and ℓ(bsm) such that (ℓ(apn), ℓ(bsm)) ∈ B. A pair (X,Y) is said to have the bounded factorization property, and denoted (X,Y) ∈ BF, if each linear continuous operator T: X → X that factors over Y is bounded. We also prove that injective tensor products of some classical Köthe spaces have bounded factorization property.