Some properties of the vector-valued Banach ideal space $E(X)$ derived from those of $E$ and $X$

Let $(\Omega,\Sigma,\mu)$ be an arbitrary measure space, $E$ a Banach idel space (Köthe function space) on $\Omega, X$ a Banach space, and $E(X) $ the "vector-valued Banach ideal space" composed of $E$ and $X$. By a general method based on semi-embeddings, it is proved for certain properties $P$ of Banach spaces that if $E$ and $X$ have $P$ then $E(X)$ has $P$ as well $(\ast)$. Examples for $P$ are the analytic Radon-Nikodým property and the property not to contain $c_0$, thus simplifying results of Bukhavlov. $(\ast)$ is also true for the type $II-\Lambda$-Randon-Nikodým property and the separable complementation property, and somewhat weaker versions of $(\ast)$ hold for the type $I-\Lambda$-Radon-Nikodým property, the property $(P)$ of costé and Lust-Piquard, and for the near Radon-Nikodým property.