The extension of p-adic compact operators

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 superset of Y, every compact operator T : Y --> X has an extension to a compact operator (T) over bar : Z --> 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.