A characterization of compact operators on ℓ^p-spaces

2025/06/07 by Mortaza Abtahi, Abtahi, Mortaza
Mathematics · #47A30 #47B37 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Primary 46B45 #Secondary 47A12

paper · pdf · doi:10.48550/arxiv.2506.06726

openalex publication_date 2025/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be a Banach space, p>1, and 1/p+1/q=1. If a sequence a=(a_i) in A has a finite p-sum, then the operator Λ_a:ℓ^q→ A, defined by Λ_a(β)=∑_i=1^∞ β_i a_i, β=(β_i)∈ ℓ^q, is compact. We present a characterization of compact operators Λ:ℓ^q→ A, and prove that Λ is compact if and only if Λ=Λ_a, for some sequence a=(a_i) in A with \(ϕ(a_i)): ϕ∈ A^*, ‖ϕ‖≤ 1\ being a totally bounded set in ℓ^p. For a sequence (T_i) of bounded operators on a Hilbert space H, the corresponding operator T:ℓ^q→ B(H), defined by T(β) = ∑_i=1^∞ β_i T_i, is compact if and only if the set \⟨ T x,x⟩:‖x‖=1\ is a totally bounded subset of ℓ^p, where ⟨ T x,x⟩ = (⟨ T₁ x,x⟩, ⟨ T₂ x,x\rangel, \dotsc), for x∈ H. Similar results are established for p=1 and p=∞.

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