  We 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 $\|\pi_B\|$ 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.

Functional Analysis