Let G be a locally compact abelian group and ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. We study Hilbert transforms associated with G-flows on ℳ and closed semigroups Σ of Ĝ satisfying the condition Σ ∪ (-Σ) = Ĝ. We prove that Hilbert transforms on such closed semigroups satisfy a weak-type estimate and can be extended as linear maps from L¹(ℳ,τ) into . 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 Im(x) belongs to the trace class then the singular values of x are O(1/n).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-1-2,
author = {Narcisse Randrianantoanina},
title = {Spectral subspaces and non-commutative Hilbert transforms},
journal = {Colloquium Mathematicae},
volume = {91},
year = {2002},
pages = {9-27},
zbl = {1004.46041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-1-2}
}
Narcisse Randrianantoanina. Spectral subspaces and non-commutative Hilbert transforms. Colloquium Mathematicae, Tome 91 (2002) pp. 9-27. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-1-2/