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PDF-Chapter14ErgodicTheoryThroughoutthischapterweassume(X;F;)isaprobabili PDF document

19414ErgodicTheoryLemma141IfTHHisalinearisometryonaHilbertspaceHthenthereisanorthogonaldirectsum145HKRwhere146KKerITKerITRRangeITandRistheclosureofRProofFirstnotethatR

19414.ErgodicTheoryLemma14.1.IfT:H!HisalinearisometryonaHilbertspaceH thenthereisanorthogonaldirectsum(14.5)H=KR;where(14.6)K=Ker(IT)=Ker(IT);R=Range(IT);andRistheclosureofR.Proof.First notethatR

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