On $\lambda$-commuting and left (right) pseudospectrum and left (right) condition pseudospectrum of continuous linear operators on ultrametric Banach spaces

In this paper, we demonstrate some spectral properties of the $\lambda$-commuting of continuous linear operators on ultrametric Banach spaces and we introduce and study the operator equations $ASB=S$ and $AS=SB.$ We give some properties of these operator equations. Some illustrative examples are provided. On the other hand, we introduce and study the left (right) pseudospectrum and the left (right) condition pseudospectrum of continuous linear operators on ultrametric Banach spaces. We prove that the left pseudospectra associated with various $\varepsilon>0$ are nested sets and the intersection of all the left pseudospectra is the left spectrum. We give a relationship between the left (right) pseudospectrum and the left (right) condition pseudospectrum. Moreover, many results are proved concerning the left (right) pseudospectrum and the left (right) condition pseudospectrum of continuous linear operators on ultrametric Banach spaces.