m at h . FA ] 4 O ct 2 00 5 WEAK OH - TYPE 2 AND WEAK OH - COTYPE 2 OF OPERATOR SPACES

Recently, OH-type and OH-cotype of operator spaces, an operator space version of type and cotype, were introduced and investigated by the author. In this paper we define weak OH-type 2 (resp. weak OH-cotype 2) of operator spaces, which lies strictly between OH-type 2 (resp. OH-cotype 2) and OH-type p for all 1 ≤ p < 2. (resp. OH-cotype q for all 2 < q ≤ ∞) This is an analogue of weak type 2 and weak cotype 2 in Banach space case, so we develop equivalent formulations of such conditions. We also consider weak operator Hilbert space, spaces with weak OH-type 2 and weak OH-cotype 2 simultaneously and establish corresponding equivalent formulations. At the end of this paper we construct an Hilbertian operator space X and a nonHilbertian operator space T , which are nontrivial examples of weak operator Hilbert space. Their construction is quite similar to that of 2-convexified Tsirelson’s space by W. B. Johnson.