Given an operator T : X → Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices T E : E(X) → E(Y) given by T E (f)(ω) = T(f(ω)). It is proved that for any Banach lattice E which does not contain c 0, the operator T is an isomorphism on a subspace isomorphic to c 0 if and only if so is T E . An analogous result for invertible operators on subspaces isomorphic to ℓ 1 is also given.
Positivity – Springer Journals
Published: Jun 8, 2010
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