New square function characterizations of operator-valued Hardy spaces on the Euclidean space \(\mathbb {R}^d\)

Let \(\mathcal {M}\) be a von Neumann algebra equipped with a normal semifinite faithful trace \(\tau \). Let \(\mathcal {H}_p(\mathbb {R}^d,\,\mathcal {M})\) denote the operator-valued Hardy space with \(1\le p<\infty \), which is first studied by T. Mei [Mem. Amer. Math. Soc. 188 (2007), vi+64 pp; MR2327840]. In this paper, the authors mainly establish some new square function characterizations of operator-valued Hardy space \(\mathcal {H}_p(\mathbb {R}^d,\,\mathcal {M})\) for all \(1\le p<\infty \), which can describe the predual spaces of noncommutative BMO spaces.