We study a factorization of bounded linear maps from an operator space to its dual space . It is shown that $T: A \rightarrow A*$ factors through a pair of column Hilbert space and its dual space if and only if is a bounded linear form on by the canonical identification equipped with a numerical radius type Haagerup norm. As a consequence, we characterize a bounded linear map from a Banach space to its dual space, which factors through a pair of Hilbert spaces.
Takashi ITOH. Masaru NAGISA. "Numerical radius Haagerup norm and square factorization through Hilbert spaces." J. Math. Soc. Japan 58 (2) 363 - 377, April, 2006. https://doi.org/10.2969/jmsj/1149166780