2015/09/08 by Lindemulder, Nick
#46B10 (secondary) #46E30 #46E40 (primary) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1509.02493
Let E and G be two Banach function spaces, let T ∈ L(E,Y), and let ⟨ X,Y ⟩ be a Banach dual pair. In this paper we give conditions for which there exists a (necessarily unique) bounded linear operator TY ∈ L(E(Y),G(Y)) with the property that ⟨ x,TYe ⟩ = T⟨ x,e ⟩, e ∈ E(Y), x ∈ X. Our first main result states that, in case ⟨ X,Y ⟩ = ⟨ Y*, Y ⟩ with Y a reflexive Banach space, for the existence of TY it sufficient that T is dominated by a positive operator. Our second main result concerns the case that T is an adjoint operator on L∞(A): we suppose that E = L∞(A) for a semi-finite measure space (A,\mathscrA,μ), that ⟨ F, G ⟩ is a Köthe dual pair, and that T is σ(L∞(A),L1(A))-to-σ(G,F) continuous. Then TY exists provided that T is dominated by a positive operator, in which case TY is σ(L∞(A;Y),L1(A;X))-to-σ(G(Y),F ⊗ X) continuous; here F ⊗ X denotes the closure of F ⊗ X in F(X). We also consider situations in which the existence is automatic and we furthermore show that in certain situations it is necessary that T is regular. As an application of this result we consider conditional expectation on Banach space-valued L∞-spaces.