Common properties of the operator equations in ultrametric specrtal theory

Let X and Y be two ultrametric Banach spaces over K. Let A,D ∈ B(X,Y) and B,C ∈ B(Y,X) such that ABA=ACA (resp. ACD=DBD and DBA=ACA). In this paper, the operator equation ABA=ACA is studied, and the common operator properties of AC-IY and BA-IX are described. In particular, it is proved that N(IY-AC) is complemented in Y if and only if N(IX-BA) is complemented in X. Moreover, the approach is generalized (i.e., CD=DBD and DBA=ACA) for considering relationships between the properties IY-AC and IX-BD. Finally, several illustrative examples are provided.