Premium
How do the positive embeddings of C 0 ( K , X ) Banach lattices depend on the αth derivatives of K?
Author(s) -
Galego Elói Medina,
RincónVillamizar Michael A.
Publication year - 2017
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600244
Subject(s) - mathematics , hausdorff space , banach space , locally compact space , simple (philosophy) , pure mathematics , vector space , isomorphism (crystallography) , extension (predicate logic) , discrete mathematics , combinatorics , crystal structure , philosophy , chemistry , epistemology , crystallography , computer science , programming language
Let K and S be locally compact Hausdorff spaces and X be an abstract L p space. Suppose that T is a positive Banach lattice isomorphism fromC O ( K )intoC O ( S , X ) . Then for each ordinal α the cardinalities of the αth derivativesK ( α )andS ( α )satisfy the following inequalityK ( α )1 / p ≤ ∥ T ∥T − 1S ( α )1 / p . Moreover, if∥ T ∥T − 1< 2 1 / p , then K ( α )is a continuous image of a subset of S ( α )which can be taken closed when K is compact. The first statement of this result for p = 1 is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number2 1 / pis sharp in these vector‐valued theorems.