Strict singularity of a Volterra-type integral operator on H^p

2015/09/28 by Santeri Miihkinen, Miihkinen, Santeri
Mathematics · #30H10 (Secondary ) #47G10 (Primary) #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA #msc:30H10 #msc:47G10

paper · pdf · doi:10.48550/arxiv.1509.08356

14 pages, 1 figure

arxiv created 2015/09/28 · openalex publication_date 2015/09/28 · arxiv updated 2015/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We prove that a Volterra-type integral operator T_gf(z) = ∫₀^z f(ζ)g'(ζ)dζ, z ∈ \mathbb D, defined on Hardy spaces H^p, 1 ≤ p < ∞, fixes an isomorphic copy of ℓ^p, if the operator T_g is not compact. In particular, this shows that the strict singularity of the operator T_g coincides with the compactness of the operator T_g on spaces H^p. As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator T_g on H¹.

Citations

• On Analytic Functions of Bounded Mean Oscillation

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• Rigidity of Volterra-type integral operators on Hardy spaces of the unit ball
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