Spectral subspaces and non-commutative Hilbert transforms
Tom 91 / 2002
Streszczenie
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 $O(1/n)$.
