Some ergodic theorems for multipliers of commutative Banach algebras

This thesis is about some ergodic properties of the multipliers of commutative Banach algebra and consists of eight chapters: Introduction, Basic Definitions-Theorems, Basic Concepts of Banach Algebras and Multipliers Theory, the Results Obtained, and Discussion-Conclusion. In the first chapter of the thesis, it is mentioned about the importance in the literature and historical development of the subject. The following three chapters of the thesis are related to some of the salient facts about Banach algebras and the theory of multipliers. In these chapters of thesis, we recall main definitions and theorems which are used through the thesis. The research findings that constitute the main part of this thesis are divided into three chapters. In this thesis, it was firstly proved that every power bounded multiplier can be extended the isometric multiplier on a Banach algebra. Using these results obtained with respect to the isometric multipliers, some necessary and sufficient conditions on convergence of iterations of multipliers in a commutative Banach algebras are given. Secondly, we concentrated on the structure of more general version of Katznelson-Tzafriri theorem. Similar results were also obtained for multipliers on Banach algebras. And lastly, theorems of Choquet-Deny type on ergodic properties of measures on locally compact abelian groups were discussed and Choquet-Deny theorem was generalized in the context of multipliers of commutative Banach algebras, in the sequel, as an application of our main results, these results were applied to the weighted group algebras and Segal algebras.