A new approach to $\gamma$-bounded representations

Abstract

Let $X$ be a Banach space, let $(\Omega ,\mu )$ be a $\sigma$-finite measure space and let $A,B:\Omega \to B(X)$ 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 $a_1\in L^{\infty }(\Omega ;K)$ and $a_2\in L^{\infty }(\Omega ;K)$ such that $⟨B(t)A(s)x,x^{\ast }⟩=(a_2(t)|a_1(s){)}_K$ for a.e. $(s,t)$ in ${\Omega }^2$, with $\Vert a_1{\Vert }_{\infty }\Vert a_2{\Vert }_{\infty }\le \gamma (A)\gamma (B)\Vert x||x^{\ast }\Vert$. This factorization property allows us to improve or simplify some results concerning $\gamma$-bounded representations of groups or semigroups.

To appear in "Operator Theory 28, Conference Proceedings, Timişoara 2022"