Regularity and <i>R</i> ‐Decomposition

ABSTRACT Let be a complex unital Banach algebra and be a regularity. In order to characterize the accumulation point of the spectrum of associated with , we define ‐decomposition of elements in , and give the conditions to make a regularity, where is the set of elements with ‐decomposition. Further, we study the relation between the spectrum of associated with and the accumulation point of the spectrum of associated with . Besides, some properties of are discussed.