Weak type $(2,H)$ and weak cotype $(2,H)$ of operator spaces

Abstract

Recently an operator space version of type and cotype, namely type $(p,H)$ and cotype $(q,H)$ of operator spaces for $1\le p\le 2\le q\le \infty$ and a subquadratic and homogeneous Hilbetian operator space $H$ were introduced and investigated by the author. In this paper we define weak type $(2,H)$ (resp. weak cotype $(2,H)$) of operator spaces, which lies strictly between type $(2,H)$ (resp. cotype $(2,H)$) and type $(p,H)$ for all $1\le p<2$ (resp. cotype $(q,H)$ for all $2<q\le \infty$). This is an analogue of weak type 2 and weak cotype 2 in the Banach space case, so we develop analogous equivalent formulations. We also consider weak-$H$ space, spaces with weak type $(2,H)$ and weak cotype $(2,H^{\ast })$ simultaneously and establish corresponding equivalent formulations.

22 pages, Definition is broadened, and eigen value estimation for weak $H$-space has been added. Tsirelson type construction is removed