On the operator equation ABA=ACA and its generalization on non-Archimedean Banach spaces

Abstract

Let $X$ and $Y$ be non-Archimedean Banach spaces over a non-Archimedean valued field $\mathbb{K}.$ In this paper, we study some properties of $A\in\mathcal{L}(X,Y)$ and $B,C\in\mathcal{L}(Y,X)$ such that $ABA=ACA$ and many basic operator properties in common of $AC-I_{Y}$ and $BA-I_{X}$ are given. In particular, $N(I_{Y}-AC)$ is a complemented subspace of $Y$ if and only if $N(I_{X}-BD)$ is a complemented subspace of $X.$ Moreover, the approach is generalized for considering relationships between the properties of $I_{Y}-AC$ and $I_{X}-BD.$ Finally, several illustrative examples are provided.