Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP

2025/06/21 by Grando, Thiago, Santos, Elisa R.
#46E40 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46B20 #secondary 46B25

paper · doi:10.48550/arxiv.2506.17762

Abstract

In this paper, we study the Bishop-Phelps-Bollobás property for operators (BPBp for short). To this end, we investigate the generalized approximate hyperplane series property (generalized AHSP for short) for a pair (X,Y) of Banach spaces, which characterizes when (ℓ₁(X),Y) has the BPBp. We prove the following results. For a locally compact Hausdorff space L, if (X, C₀(L,Y)) has the BPBp, then so does (X,Y). Furthermore, if the pair (X, Y) has the generalized AHSP and L(X,Z) = K(X,Z), then the pair (X, Z) also has the generalized AHSP, where Z is one of the spaces C(K, Y), C₀(L, Y), or C_b(Ω, Y), with K a compact Hausdorff space and Ω a completely regular space.

Citations

• The Bishop-Phelps-Bollobás property for compact operators

Related

• The Bishop-Phelps-Bollobás point property
• The Bishop-Phelps-Bollobas property for certain Banach spaces
• The strong Bishop-Phelps-Bollobás property
• The Bishop-Phelps-Bollobás property for operators defined on c₀-sum of Euclidean spaces
• Bishop-Phelps-Bolloba's theorem on bounded closed convex sets