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The structure of compact disjointness preserving operators on continuous functions
Author(s) -
Lin YingFen,
Wong NgaiChing
Publication year - 2009
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.200610786
Subject(s) - mathematics , disjoint sets , hausdorff space , sequence (biology) , operator (biology) , operator norm , combinatorics , norm (philosophy) , limit of a sequence , discrete mathematics , countable set , graph , compact operator , pure mathematics , extension (predicate logic) , operator theory , mathematical analysis , biochemistry , chemistry , genetics , repressor , limit (mathematics) , gene , transcription factor , political science , computer science , law , biology , programming language
Let T be a compact disjointness preserving linear operator from C 0 ( X ) into C 0 ( Y ), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Σ nδ x n⊗ h n for a (possibly finite) sequence { x n } n of distinct points in X and a norm null sequence { h n } n of mutually disjoint functions in C 0 ( Y ). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)