On the τ-Compactness of Products of τ -Measurable Operators Adjoint to Semi-Finite Von Neumann Algebras

Abstract

Let \( \mathcal{M} \) be the von Neumann algebra of operators in a Hilbert space \( \mathcal{H} \) and τ be an exact normal semi-finite trace on \( \mathcal{M} \). We obtain inequalities for permutations of products of τ-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and Pólya) of products of τ -measurable operators and a sufficient condition of orthogonality of certain nonnegative τ-measurable operators. We state sufficient conditions of the τ –compactness of products of self-adjoint τ -measurable operators and obtain a criterion of the τ -compactness of the product of a nonnegative τ-measurable operator and an arbitrary τ -measurable operator. We present an example that shows that the nonnegativity of one of the 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 τ = tr.