## Abstract

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.

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 language | English |
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Title of host publication | Ordered Structures and Applications |

Subtitle of host publication | Positivity VII |

Editors | Marcel de Jeu, Ben de Pagter, Onno van Gaans, Mark Veraar |

Place of Publication | Cham |

Publisher | Birkhäuser |

Pages | 281-306 |

Number of pages | 26 |

ISBN (Electronic) | 978-3-319-27842-1 |

ISBN (Print) | 978-3-319-27840-7 |

DOIs | |

Publication status | Published - 2016 |

### Publication series

Name | Trens in Mathematics |
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ISSN (Print) | 2297-0215 |

## Keywords

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