Noncommutative Hardy–Lorentz spaces associated with semifinite subdiagonal algebras

Abstract

Let $\mathcal{A}$ be a maximal subdiagonal algebra of semifinite von Neumann algebra $\mathcal{M}$. For $0<p\le \infty$, we define the noncommutative Hardy–Lorentz spaces $H^{p,\omega }\left(\mathcal{A}\right)$, and give some properties of these spaces. We obtain that the Herglotz maps are bounded linear maps from ${\Lambda }_{\omega }^p\left(\mathcal{M}\right)$ into ${\Lambda }_{\omega }^p\left(\mathcal{M}\right)$, and with this result we characterize the dual spaces of $H^{p,\omega }\left(\mathcal{A}\right)$ for $1<p<\infty$. We also present the Hartman–Wintner spectral inclusion theorem of Toeplitz operators and Pisier’s interpolation theorem for this case.