On the operator equations ABA = A2 and BAB = B2 on non-Archimedean Banach spaces

Abstract

Let XX and YY 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 ABA=A2ABA={A}^{2} and BAB=B2.BAB={B}^{2}. In this article, we investigate some properties of the operator equations ABA=A2ABA={A}^{2} and BAB=B2BAB={B}^{2}, and many common basic properties of IY−AB{I}_{Y}-AB and IX−BA{I}_{X}-BA are given. In particular, if XX and YY are Banach spaces over a spherically complete field K,{\mathbb{K}}, then N(IY−AB)N\left({I}_{Y}-AB) is a complemented subspace of YY if and only if N(IX−BA)N\left({I}_{X}-BA) is a complemented subspace of X.X. Finally, we give some examples to illustrate our work.

Keywords

• non-archimedean banach spaces
• operator equation
• operators theory
• 47a10
• 47s10
• 46s10