Spectral subspaces and non-commutative Hilbert transforms

Narcisse Randrianantoanina¹

¹ Department of Mathematics and Statistics Miami University Oxford, OH 45056, U.S.A.

Colloquium Mathematicum, Tome 91 (2002) no. 1, pp. 9-27

Résumé

Let $G$ be a locally compact abelian group and ${\mathcal M}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau $. We study Hilbert transforms associated with $G$-flows on ${\mathcal M}$ and closed semigroups ${\mit\Sigma }$ of $\widehat G$ satisfying the condition ${\mit\Sigma } \cup (-{\mit\Sigma })=\widehat {G}$. We prove that Hilbert transforms on such closed semigroups satisfy a weak-type estimate and can be extended as linear maps from $L^1({\mathcal M},\tau )$ into $L^{1,\infty }({\mathcal M}, \tau )$. As an application, we obtain a Matsaev-type result for $p=1$: if $x$ is a quasi-nilpotent compact operator on a Hilbert space and $\mathop {\rm Im}\nolimits (x)$ belongs to the trace class then the singular values $\{\mu _n(x)\}_{n=1}^\infty $ of $x$ are $O(1/n)$.

Détail

DOI : 10.4064/cm91-1-2

Keywords: locally compact abelian group mathcal semifinite von neumann algebra faithful semifinite normal trace tau study hilbert transforms associated g flows mathcal closed semigroups mit sigma widehat satisfying condition mit sigma cup mit sigma widehat prove hilbert transforms closed semigroups satisfy weak type estimate extended linear maps mathcal tau infty mathcal tau application obtain matsaev type result quasi nilpotent compact operator hilbert space mathop nolimits belongs trace class singular values infty