Spectral subspaces and non-commutative Hilbert transforms
Volume 91 / 2002
Abstract
Let 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).