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# Journal of Operator Theory

Volume 63, Issue 2, Spring 2010  pp. 333-347.

Summary:  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.