On spaces of compact operators on $C(K, X)$ spaces

Abstract:

This paper concerns the spaces of compact operators ${\mathcal K}(E, F)$, where $E$ and $F$ are Banach spaces $C([1, \xi ], X)$ of all continuous $X$-valued functions defined on the interval of ordinals $[1, \xi ]$ and equipped with the supremun norm. We provide sufficient conditions on $X$, $Y$, $\alpha$, $\beta$, $\xi$ and $\eta$, with $\omega \leq \alpha \leq \beta < \omega _{1}$ for the following equivalence:

1. [(a)] ${\mathcal K}(C([1, \xi ], X), C([1, \alpha ], Y))$ is isomorphic to ${\mathcal K} (C([1, \eta ], X), C([1, \beta ], Y))$,

2. [(b)] $\beta < \alpha ^{\omega }$.

In this way, we unify and extend results due to Bessaga and Pełczyński (1960) and C. Samuel (2009). Our result covers the case of the classical spaces $X=l_{p}$ and $Y=l_{q}$, with $1<p, q< \infty$.