Factorization in Fréchet spaces
Consider the following conditions. (a) Every regular LB-space is complete; (b) if an operator T between complete LB-spaces maps bounded sets into relatively compact sets, then T factorizes through a Montel LB-space; (c) for every complete LB-space E the space C (βℕ, E) is bornological. We show that (a) ⇒ (b) ⇒ (c). Moreover, we show that if E is Montel, then (c) holds. An example of an LB-space E with a strictly increasing transfinite sequence of its Mackey derivatives is given.
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...
Starting with a continuous injection I: X → Y between Banach spaces, we are interested in the Fréchet (non Banach) space obtained as the reduced projective limit of the real interpolation spaces. We study relationships among the pertenence of I to an operator ideal and the pertenence of the given interpolation space to the Grothendieck class generated by that ideal.
We characterize all Fréchet quotients of the space (Ω) of (complex-valued) real-analytic functions on an arbitrary open set . We also characterize those Fréchet spaces E such that every short exact sequence of the form 0 → E → X → (Ω) → 0 splits.
Fréchet spaces of strongly, weakly and weak*-continuous Fréchet space valued functions are considered. Complete solutions are given to the problems of their injectivity or embeddability as complemented subspaces in dual Fréchet spaces.