THE EXTENSION OF $p$-ADIC
COMPACT OPERATORS

N. DE GRANDE-DE KIMPE
AND C. PEREZ-GARCIA

Abstract:

This paper is devoted to study the non-Archimedean locally convex spaces $X$ having the following property: For all non-Archimedean locally convex spaces $Z\supset Y$, every compact operator $T:Y\to X$ has an extension to a compact operator ${\overline T}:Z\to X$. The results obtained depend strongly on the spherical completeness of the ground field. On the other hand, the situation here is completely different from its Archimedean counterpart. Our results also lead to some new characterizations of spherically complete fields and of discretely valued fields.