On spaces of operators on $C(Q)$ spaces ($Q$ countable metric space)

In this paper we study spaces of nuclear operators ${\mathcal N}(C(Q))$ and spaces of compact operators ${\mathcal K}(C(Q))$ on spaces of continuous functions $C(Q),$ where $Q$ is a countable compact metric space, in connection with the C. Bessaga and A. Pełczyński isomorphic classification of these spaces.

We show that the spaces ${\mathcal K}(C(Q))$ [resp. ${\mathcal N}(C(Q))$] and ${\mathcal K}(C(Q'))$ [resp. ${\mathcal N}(C(Q'))$] are isomorphic if, and only if, $C(Q)$ and $C(Q')$ are isomorphic. We show also that ${\mathcal N}(C(Q))$ is not isomorphic to a subspace of ${\mathcal K}(C(Q)).$