On isomorphic classifications of spaces of compact operators

We prove an extension of the classical isomorphic classification of Banach spaces of continuous functions on ordinals. As a consequence, we give complete isomorphic classifications of some Banach spaces ${\mathcal K}(X, Y^{\eta})$, $\eta \geq \omega$, of compact operators from $X$ to $Y^{\eta}$, the space of all continuous $Y$-valued functions defined in the interval of ordinals $[1, \eta]$ and equipped with the supremum norm. In particular, under the Continuum Hypothesis, we extend a recent result of C. Samuel by classifying, up to isomorphism, the spaces ${\mathcal K}(X^{\xi}, c_{0}(\Gamma)^{\eta})$, where $\omega \leq \xi < \omega_1$, $\eta \geq \omega$, $\Gamma$ is a countable set, $X$ contains no complemented copy of $l_1$, $X^*$ has the Mazur property and the density character of $X^{**}$ is less than or equal to $\aleph_{1}$.