Bounded factorization property for Fréchet spaces

Abstract

An operator T ∈ L(E, F) factors over G if T = RS for some S ∈ L(E, G) and R ∈ L(G, F); the set of such operators is denoted by LG(E, F). A triple (E, G, F) satisfies bounded factorization property (shortly, (E, G, F) ∈ ℬ︁ℱ) if LG(E, F) ⊂ LB(E, F), where LB(E, F) is the set of all bounded linear operators from E to F. The relationship (E, G, F) ∈ ℬ︁ℱ is characterized in the spirit of Vogt's characterisation of the relationship L(E, F) = LB(E, F) [23]. For triples of K�othe spaces the property ℬ︁ℱ is characterized in terms of their K�othe matrices. As an application we prove that in certain cases the relations L(E, G1) = LB(E, G1) and L(G2, F) = LB(G2, F) imply (E, G, F) ∈ ℬ︁ℱ where G is a tensor product of G1 and G2.