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On order preserving contractions
Authors:Foguel  S R
Institution:(1) The Hebrew University of Jerusalem, Israel
Abstract:Let (Ω,Σ,μ) be a measure space and letP be an operator onL 2(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P n x||=||P *n x||=||x||,n=1,2,...} thenK=L 2(Ω,Σ1,μ), where Σ1 ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication. This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.
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