Bounded Operators and Complemented Subspaces of Cartesian Products

Abstract

We study the structure of complemented subspaces in Cartesian products X × Y of Köthe spaces X and Y under the assumption that every linear continuous operator from X to Y is bounded. In particular, it is proved that each non-Montel complemented subspace with absolute basis E ⊂ X × Y is isomorphic to a space of the form E1 × E2 , where E1 is a complemented subspace of X and E2 is a complemented subspace of Y.

On Orlicz-Power Series Spaces

“…It can be derived (see, e.g., [4], Proposition 14) from the general characterization of the property (X, Y ) ∈ B (Vogt [17]). …”

A study on certain Köthe spaces

“…The spaces λ p (A), 1 < p ≤ ∞ are called as generalized Köthe spaces by Bierstedt et al [1]. In some sources, for example [3,7], the spaces λ p (A) denoted by K p (A) and called by p −Köthe space for 1 ≤ p < ∞.…”

“…Equipped with semi-norms given by (1) K (A) is a Fréchet space, [3]. It is well-known that the space bs of bounded series is defined by…”

“…and is an admissible space with the norm • bs . Following [3,7], we define the new space λ bs (A) by…”