2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160182We consider BMO spaces of operator-valued functions, among them the space of operator-valued functions $B$ which define a bounded paraproduct on $L^2(H)$. We obtain several equivalent formulations of $\|π_B\|$ in terms of the norm of the "sweep" function of $B$ or of averages of the norms of martingales transforms of $B$ in related spaces. Furthermore, we investigate a connection between John-Nirenberg type inequalities and Carleson-type inequalities via a product formula for paraproducts and deduce sharp dimensional estimates for John-Nirenberg type inequalities.to appear in J. Operator TheoryFunctional Analysis42B30, 42B35 (Primary); 47B35 (Secondary)text