Optimal extensions for p-th power factorable operators

2015/11/07 by O. Delgado, Delgado, O., E. A. Sanchez Perez +1
Mathematics · #46E30 #46G10 #47B38 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46E30 #msc:46G10 #msc:47B38

paper · pdf · doi:10.48550/arxiv.1511.02335

arxiv created 2015/11/07 · arxiv updated 2015/11/10

Abstract

Let X(μ) be a function space related to a measure space (Ω,Σ,μ) with χ_Ω∈ X(μ) and let T\colon X(μ)→ E be a Banach space valued operator. It is known that if T is p-th power factorable then the largest function space to which T can be extended preserving p-th power factorability is given by the space L^p(m_T) of p-integrable functions with respect to m_T, where m_T\colonΣ→ E is the vector measure associated to T via m_T(A)=T(χ_A). In this paper we extend this result by removing the restriction χ_Ω∈ X(μ). In this general case, by considering m_T defined on a certain δ-ring, we show that the optimal domain for T is the space L^p(m_T)∩ L¹(m_T). We apply the obtained results to the particular case when T is a map between sequence spaces defined by an infinite matrix.

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