The Bishop-Phelps-Bollobás property for compact operators

2016/04/03 by Dantas, Sheldon, Garcia, Domingo, Maestre, Manuel +1 · 2 citations
#46B25 #46B28 #46E40 #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 46B04 #Secondary: 46B20

paper · doi:10.48550/arxiv.1604.00618

Abstract

We study the Bishop-Phelps-Bollobás property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let X, Y be Banach spaces. If (c₀,Y) has the BPBp for compact operators, then so do (C₀(L),Y) for every locally compact Hausdorff topological space L and (X,Y) whenever X^* is isometrically isomorphic to ℓ₁. If X^* has the Radon-Nikodým property and (ℓ₁(X),Y) has the BPBp for compact operators, then so does (L₁(μ,X),Y) for every positive measure μ; as a consequence, (L₁(μ,X),Y) has the the BPBp for compact operators when X and Y are finite-dimensional or Y is a Hilbert space and X=c₀ or X=L_p(ν) for any positive measure ν and 1< p< ∞. For 1\leqslant p

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• Bollobás-type theorems for range strongly exposing operators
• Stability results of the Bishop-Phelps-Bollobás property and the generalized AHSP

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