The Hankel operators and noncommutative BMO spaces

Abstract

Let $\mathcal{M}$ be a von Neumann algebra with a faithful normal semifinite trace $\tau$. The noncommutative Hardy space $H^p\left(\mathcal{M}\right)$ associates with $\mathcal{A}$, which is a subdiagonal algebra of $\mathcal{M}$. We define the Hankel operator $H_t$ on $H^p\left(\mathcal{M}\right)$, and we obtain that the norm $\Vert H_t\Vert$ is equal to $d(t;\mathcal{A})$ and is also the equivalent of the $BMO\left({\mathcal{M}}^{\mathrm{sa}}\right)$ norm of $t$ for every $t\in \mathcal{M}$, where ${\mathcal{M}}^{\mathrm{sa}}$ are the self-adjoint operators in $\mathcal{M}$.