Some mean ergodic theorems on locally compact hypergroups

In this paper, we study the mean ergodic theorems and the weighted ones on locally compact hypergroups. Among other obtained results, for the class of all commutative hypergroups \mathcal{H} with a Plancherel measure \widetilde{\omega} that {\rm supp}(\widetilde{\omega})=\widehat{\mathcal{H}}, we prove that if \left(k_{j}\right)_{j \in \mathbb{N}} is a subsequence of \mathbb{N}, f\in L^۲(\mathcal{H}), and \mu is a power bounded measure on \mathcal{H} such that the sequence\left(\frac{۱}{m} \sum_{n=۱}^{m}\underbrace{\mu\ast\ldots\ast\mu}_{k_{n}-\text{times}}\ast f\right)_{m\in\mathbb{N}}weakly converges in L^۲(\mathcal{H}), then the numerical sequence \left(\frac{۱}{m} \sum_{n=۱}^{m} \alpha^{k_n}\right)_{m\in\mathbb{N}} is convergent too for all \alpha\in \mathbb{C} with \tilde{\omega}\left(\{\xi\in\widehat{\mathcal{H}}:\hat{\mu}(\xi)=\alpha\}\right)>۰.

کلیدواژه ها:

locally compact hypergroups ، mean ergodic theorem ، power bounded measures ، {\rm w}^*-convergence ، Fourier transforms