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

Abstract

Let M be the von Neumann algebra of operators in a Hilbert space H and $\tau$ be an exact normal semi-finite trace on 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 P´olya) 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 the factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from M. All results are new for the *-algebra B(H) of all bounded linear operators in H endowed with the canonical trace $\tau$ = tr.