A new approach to $\gamma$-bounded representations

Let $X$ be a Banach space, let $(\mathrm{\Omega },\mu )$ be a $\sigma$-finite measure space and let $A,B\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to B(X)$ be strongly measurable $\gamma$-bounded functions. We show that for all $x\in X$ and all $x^{\ast }\in X^{\ast }$, there exist a Hilbert space $K$ and two measurable functions $a_1\in L^{\mathrm{\infty }}(\mathrm{\Omega };K)$ and $a_2\in L^{\mathrm{\infty }}(\mathrm{\Omega };K)$ such that $⟨B(t)A(s)x,x^{\ast }⟩=(a_2(t)\mathrm{\mid }{a}_1(s){)}_K$ for a.e. $(s,t)$ in ${\mathrm{\Omega }}^2$, with $\mathrm{\parallel }{a}_1{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{a}_2{\mathrm{\parallel }}_{\mathrm{\infty }}\le \gamma (A)\gamma (B)\mathrm{\parallel }x\mathrm{\mid }\mathrm{\mid }{x}^{\ast }\mathrm{\parallel }$. This factorization property allows us to improve or simplify some results concerning $\gamma$-bounded representations of groups or semigroups.