  Generalizing Pisier's idea, we introduce a Hilbertian matrix cross normed
space associated with a pair of symmetric normed ideals. When the two ideals
coincide, we show that our construction gives an operator space if and only if
the ideal is the Schatten class. In general, a pair of symmetric normed ideals
that are not necessarily the Schatten class may give rise to an operator space.
We study the space of completely bounded mappings between the matrix cross
normed spaces obtained in this way and show that the multiplicator norm
naturally appears as the completely bounded norm.
