Krengel-Lin decomposition for probability measures on hypergroups |
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Authors: | CRobinson Edward Raja |
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Institution: | Stat-Math Unit, Indian Statistical Insitute, 8th Mile Mysore Road, R.V. College Post, Bangalore 560 059, India |
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Abstract: | A Markov operator P on a σ-finite measure space (X,Σ,m) with invariant measure m is said to have Krengel-Lin decomposition if L2(X)=E0⊕L2(X,Σd) where E0={f∈L2(X)∣‖Pn(f)‖→0} and Σd is the deterministic σ-field of P. We consider convolution operators and we show that a measure λ on a hypergroup has Krengel-Lin decomposition if and only if the sequence converges to an idempotent or λ is scattered. We verify this condition for probabilities on Tortrat groups, on commutative hypergroups and on central hypergroups. We give a counter-example to show that the decomposition is not true for measures on discrete hypergroups. |
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Keywords: | 43A62 60B99 |
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