Characterization of singular numbers of products of operators in matrix algebras and finite von Neumann algebras

TL;DR

This paper characterizes the possible generalized singular numbers of operator products in finite von Neumann algebras and matrix algebras, providing new inequalities without invertibility assumptions.

Contribution

It introduces a novel characterization of singular numbers for products of operators, extending previous results to non-invertible cases in matrix and von Neumann algebras.

Findings

Derived inequalities for generalized singular numbers of operator products

Extended characterization to non-invertible operators in matrix algebras

Provided new insights into operator product singular value behavior

Abstract

We characterize in terms of inequalities the possible generalized singular numbers of a product AB of operators A and B having given generalized singular numbers, in an arbitrary finite von Neumann algebra. We also solve the analogous problem in matrix algebras M_n(C), which seems to be new insofar as we do not require A and B to be invertible.