Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Abstract

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E → F are symmetric sequence spaces with E = 1 and F = c 0 and ∞ then there exist a intermediate symmetric sequence space G such that E → G → F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c 0 and 1 inside the class of all symmetric sequence spaces are given.

Strictly singular inclusions into $$L^1 + L^\infty$$

“…From Proposition 3.5 we deduce that t α ∈ C m ϕ for some 0 < α ≤ 1 and each m ∈ N. Now, using(13), for any m ∈ N there exists a function ω m in the convex hull of {ϕ(t/n)/ϕ(1/n) : n ∈ N} such thatt α ≤ 3 ω m (t)for every t ∈ 1 m , m .…”

“…Kalton [15], [19,Theorem 4.a.10], García del Amo et al [8], Novikov [24], Astashkin [1] and the authors [13]). …”

Interpolation Banach lattices containing no isomorphic copies of c0

“…We also note that two different proofs of the result in case of symmetric sequence spaces can be found in [11].…”

“…An alternative proof of the above corollary in case of symmetric sequence spaces E is presented in [11].…”

Embeddings between Lorentz sequence spaces are strictly but not finitely strictly singular

“…When we consider weighted sequence spaces d(ω, p), with a weight ω = ω(j), equipped with a quasi-norm For the proof and more details about this startling result see the paper by Hernández, Sánchez and Semenov [4,Theorem 3.1].…”