Krengel–Lin decomposition for probability measures on hypergroups

AbstractA 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 (λ̌n∗λn) 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