Bounded operators and isomorphisms of Cartesian products of Fréchet spaces.

Bounded Operators and Complemented Subspaces of Cartesian Products

“…Indeed, since C is a Banach space, by Proposition 3 in 5, each of the spaces B , C is either finite dimensional or isomorphic to ℓ p . If both of them are finite dimensional we obtain that Q ( X ) ≃ P ( X ) ( s ) for some integer s .…”

“…We are going to prove that if X , Y are Köthe spaces with $(X,Y) \in {\cal B} $ and both X and Y have the SPP, then X × Y has the SPP also. The proof is based on the results from the previous section, and on our Proposition 3 from 5, which we recall below.…”

“…Proposition 5.2 (Prop. 3 from 5) Let X be an ℓ p ‐Köthe space, 1 ⩽ p < ∞, and let F and G be complementary subspaces in X (i.e., X = F ⊕ G ). If G is an infinite‐dimensional Banach space then G ≃ ℓ p and, moreover, F and G are isomorphic to some coordinate subspaces of X .…”

“…The following fact is crucial: If T is a bounded operator acting in a Köthe space X , then for each ε > 0 there exist complementary coordinate subspaces B and E such that B is a Banach space and the spectrum of the operator π E Ti E is situated inside the disc of center 0 and radius ε. The same fact is used in 5 to explain that a bounded perturbation of an authomorphism of a Köthe space remains isomorphism on a complement of some Banach coordinate subspace, and to modify the generalized Douady Lemma (15, 16) for pairs of Köthe spaces X and Y with the bounded factorization property (i.e., each continuous linear operator T : X → X which factors through Y is bounded). From a technical point of view we modify the method of I. Edelstein and P. Wojtaszcyk 3 for characterizing the complemented subspaces of Cartesian products of Banach spaces.…”

Bounded factorization property for Fréchet spaces

“…The last property was used in [6] to construct non-trivial pairs of Fréchet spaces (E, G) with the property (E, G, E) ∈ BF.…”