Common properties of the operator equations in ultrametric specrtal theory

Abstract

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-I_Y and BA-I_X are described. In particular, it is proved that N(I_Y-AC) is complemented in Y if and only if N(I_X-BA) is complemented in X. Moreover, the approach is generalized (i.e., CD=DBD and DBA=ACA) for considering relationships between the properties I_Y-AC and I_X-BD. Finally, several illustrative examples are provided.