Banach Space-valued Extensions of Linear Operators on L∞

Research output: Chapter in Book/Conference proceedings/Edited volumeChapterScientific


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.
The 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. We furthermore show that for Y within a wide class of Banach spaces (including the Banach lattices) the validity of this extension result for E = l∞ and G = K even characterizes the reflexivity of Y . The second main result concerns the case that T is an adjoint operator on L∞(A): we assume that E = L∞(A) for a semi-finite measure space (A, A, μ), that ⟨F,G⟩ is a Köthe dual pair, and that T is σ(L ∞ (A),L1(A))- to-σ(G, F) continuous. In this situation we show that TY also exists provided that T is dominated by a positive operator. As an application of this result we consider conditional expectation on Banach space-valued L ∞ -spaces.
Original languageEnglish
Title of host publicationOrdered Structures and Applications
Subtitle of host publicationPositivity VII
EditorsMarcel de Jeu, Ben de Pagter, Onno van Gaans, Mark Veraar
Place of PublicationCham
Number of pages26
ISBN (Electronic)978-3-319-27842-1
ISBN (Print)978-3-319-27840-7
Publication statusPublished - 2016

Publication series

NameTrens in Mathematics
ISSN (Print)2297-0215


  • Adjoint operator
  • Banach function space
  • Banach limit
  • conditional expectation
  • domination, dual pair
  • L∞
  • positive operator
  • vector-valued extension
  • reflexivity
  • Schauder basis


Dive into the research topics of 'Banach Space-valued Extensions of Linear Operators on L∞'. Together they form a unique fingerprint.

Cite this