Spectral subspaces and non-commutative Hilbert transforms

Narcisse Randrianantoanina. "Spectral subspaces and non-commutative Hilbert transforms." Colloquium Mathematicae 91.1 (2002): 9-27. <http://eudml.org/doc/283729>.

@article{NarcisseRandrianantoanina2002,
abstract = {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 $L^\{1,∞\}(ℳ, τ)$. 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 $\{μₙ(x)\}_\{n=1\}^\{∞\}$ of x are O(1/n).},
author = {Narcisse Randrianantoanina},
journal = {Colloquium Mathematicae},
keywords = {von Neumann algebras; Riesz projections; Hilbert transforms; semifinite von Neumann algebra; faithful semifinite normal trace; -flows; non-commutative weak-type estimate},
language = {eng},
number = {1},
pages = {9-27},
title = {Spectral subspaces and non-commutative Hilbert transforms},
url = {http://eudml.org/doc/283729},
volume = {91},
year = {2002},
}

TY - JOUR
AU - Narcisse Randrianantoanina
TI - Spectral subspaces and non-commutative Hilbert transforms
JO - Colloquium Mathematicae
PY - 2002
VL - 91
IS - 1
SP - 9
EP - 27
AB - 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 $L^{1,∞}(ℳ, τ)$. 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 ${μₙ(x)}_{n=1}^{∞}$ of x are O(1/n).
LA - eng
KW - von Neumann algebras; Riesz projections; Hilbert transforms; semifinite von Neumann algebra; faithful semifinite normal trace; -flows; non-commutative weak-type estimate
UR - http://eudml.org/doc/283729
ER -