On the $\tau$-compactness of products of $\tau$-measurable operators adjoint to semi-finite von Neumann algebras

A. M. Bikchentaev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Mathematical physics, Tome 140 (2017), pp. 78-87

Résumé

Let ${\mathcal M}$ be the von Neumann algebra of operators in a Hilbert space $\mathcal H$ and $\tau$ be an exact normal semi-finite trace on $\mathcal{M}$. We obtain inequalities for permutations of products of $\tau$-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and Pólya) of products of $\tau$-measurable operators and a sufficient condition of orthogonality of certain nonnegative $\tau$-measurable operators. We state sufficient conditions of the $\tau$-compactness of products of self-adjoint $\tau$-measurable operators and obtain a criterion of the $\tau$-compactness of the product of a nonnegative $\tau$-measurable operator and an arbitrary $\tau$-measurable operator. We present an example that shows that the nonnegativity of one of factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from $\mathcal{M}$. All results are new for the *-algebra $\mathcal{B}(\mathcal{H})$ of all bounded linear operators in $\mathcal{H}$ endowed with the canonical trace $\tau=\operatorname{tr}$.

Détail

Keywords: Hilbert space, linear operator, von Neumann algebra, normal semi-finite trace, $\tau$-measurable operator, $\tau$-compact operator, elementary operator, nilpotent, submajorization.
Mots-clés : permutation