On the operator equations ABA = A 2 and BAB = B 2 on non-Archimedean Banach spaces

Abstract

Abstract Let X X and Y Y be non-Archimedean Banach spaces over K {\mathbb{K}} , A ∈ B ( X , Y ) A\in B\left(X,Y) and B ∈ B ( Y , X ) B\in B\left(Y,X) such that A B A = A 2 ABA={A}^{2} and B A B = B 2 . BAB={B}^{2}. In this article, we investigate some properties of the operator equations A B A = A 2 ABA={A}^{2} and B A B = B 2 BAB={B}^{2} , and many common basic properties of I Y − A B {I}_{Y}-AB and I X − B A {I}_{X}-BA are given. In particular, if X X and Y Y are Banach spaces over a spherically complete field K , {\mathbb{K}}, then N ( I Y − A B ) N\left({I}_{Y}-AB) is a complemented subspace of Y Y if and only if N ( I X − B A ) N\left({I}_{X}-BA) is a complemented subspace of X . X. Finally, we give some examples to illustrate our work.