Extension of linear operators and Lipschitz maps into C(K)-spaces

摘要

We study the extension of linear operators with range in a C(K)-space, comparing andcontrasting our results with the corresponding results for thenonlinear problem of extending Lipschitz maps with values in aC(K)-space. We give necessary and sufficient conditions on aseparable Banach space X which ensure that every operatorT:E→C(K) defined on a subspace may be extended to an operatorilde T:X→C(K) with ∥ilde T∥≦ (1+ε)∥T∥ (forany ε0). Based on these we give new examples of suchspaces (including all Orlicz sequence spaces with separable dualfor a certain equivalent norm). We answer a question of Johnsonand Zippin by showing that if E is a weak*-closed subspace ofℓ1 then every operator T:E→C(K) can be extended to anoperator ilde T:ℓ1→C(K) with ∥ilde T∥≦(1+ε)∥T∥. We then show that ℓ1 has a universalextension property: if X is a separable Banach space containingℓ1 then any operator T:ℓ1→C(K) can be extended toan operator ilde T:X→ C(K) with ∥ilde T∥≦(1+ε)∥T∥; this answers a question of Speegle.