Krengel–Lin decomposition for probability measures on hypergroups

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 L 2( X)= E 0⊕ L 2( X, Σ d ) where E 0={ f∈ L 2( X)∣‖ P n ( f)‖→0} and Σ d 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 ( λ ̌ n∗λ n) 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.