2023/12/12 by Ioana Ghenciu, Ghenciu, Ioana, Roxana Popescu +1
Mathematics · #46B25 #46B45 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results
paper · pdf · doi:10.48550/arxiv.2312.07154
openalex publication_date 2023/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T:C(K,X)→ Y is a strongly bounded operator with representing measure m:Σ→ L(X,Y). We show that if T: B(K, X) → Y is its extension, then T is weak Dunford-Pettis (resp. weak^* Dunford-Pettis, weak p-convergent, weak^* p-convergent) if and only if T has the same property. We prove that if T:C(K,X)→ Y is strongly bounded limited completely continuous (resp. limited p-convergent), then m(A):X→ Y is limited completely continuous (resp. limited p-convergent) for each A∈ Σ. We also prove that the above implications become equivalences when K is a dispersed compact Hausdorff space.