 

N. J. Kalton
Extension of linear operators and Lipschitz maps into C(K)spaces


Published: 
September 10, 2007

Keywords: 
Banach spaces, spaces of continuous functions, Lipschitz extensions, linear extensions 
Subject: 
Primary: 46B03, 46B20 


Abstract
We study the extension of linear operators with range in a C(K)space, comparing and
contrasting our results with the corresponding results for the
nonlinear problem of extending Lipschitz maps with values in a
C(K)space. We give necessary and sufficient conditions on a
separable Banach space X which ensure that every operator
T:E→C(K) defined on a subspace may be extended to an operator
\tilde T:X→C(K) with ∥\tilde T∥≦ (1+ε)∥T∥ (for
any ε>0). Based on these we give new examples of such
spaces (including all Orlicz sequence spaces with separable dual
for a certain equivalent norm). We answer a question of Johnson
and Zippin by showing that if E is a weak*closed subspace of
ℓ_{1} then every operator T:E→C(K) can be extended to an
operator \tilde T:ℓ_{1}→C(K) with ∥\tilde T∥≦
(1+ε)∥T∥. We then show that ℓ_{1} has a universal
extension property: if X is a separable Banach space containing
ℓ_{1} then any operator T:ℓ_{1}→C(K) can be extended to
an operator \tilde T:X→ C(K) with ∥\tilde T∥≦
(1+ε)∥T∥; this answers a question of Speegle.


Acknowledgements
The author was supported by NSF grant DMS0555670


Author information
Department of Mathematics, University of MissouriColumbia, Columbia, MO 65211
nigel@math.missouri.edu

