Superprojective Banach spaces

On Subprojectivity and Superprojectivity of Banach Spaces

“…The following result [20,Proposition 3.3] is useful to show that some spaces fail subprojectivity or superprojectivity. Proposition 2.1.…”

“…Basic examples of subprojective spaces are ℓ p for 1 ≤ p < ∞ and L p (0, 1) for 2 ≤ p < ∞ [18, Proposition 2.4]; and C(K) spaces with K a scattered compact are both subprojective and superprojective [18,Propositions 2.4 and 3.4]. Moreover, recent systematic studies of subprojective spaces [28] (see also [13]) and superprojective spaces [20] have widely increased the family of known examples in those classes.…”

“…A reflexive space is subprojective (superprojective) if and only if its dual is superprojective (subprojective). In general, however, X being subprojective does not imply that X * is superprojective, and X * being subprojective does not imply that X is superprojective, and it is unknown whether the remaining implications are valid [20,Introduction]. Basic examples of subprojective spaces are ℓ p for 1 ≤ p < ∞ and L p (0, 1) for 2 ≤ p < ∞ [18, Proposition 2.4]; and C(K) spaces with K a scattered compact are both subprojective and superprojective [18,Propositions 2.4 and 3.4].…”

“…The dual space of S ⊗π S can be identified with L(S, S * ). By[20, Corollary 3.5] it is enough to show that that there is a non-compact operator in L(S, S * ). Given x = (x i ) i∈N ∈ S, we denote the decreasing rearrangement of (|x i |) i∈N by x d = (x d i ) i∈N .…”

The perturbation classes problem for subprojective and superprojective Banach spaces

“…In Theorems 5 and 7, we prove that the perturbation classes problem for Φ + (X, Y ) has a positive answer when X is subprojective and similarly that the perturbation classes problem for Φ − (X, Y ) has a positive answer when Y is superprojective. A Banach space X is subprojective if every closed infinite-dimensional subspace of X contains an infinite-dimensional subspace complemented in X; a Banach space X is superprojective if every closed infinite-codimensional subspace of X is contained in an infinite-codimensional subspace complemented in X. Subprojective and superprojective spaces were introduced by Whitley to study strictly singular and strictly cosingular operators [16]; see [7] for a fairly complete list of examples known at the time and [11] [8] and [4] for more recent discoveries.…”