Factorization of positive-semidefinite operators with absolutely summable entries

TL;DR

This paper investigates the factorization of positive-semidefinite operators with absolutely summable entries, providing finite-dimensional analysis, duality results, and conditions for infinite-dimensional decompositions.

Contribution

It offers a new linear programming reformulation, duality analysis, and characterizes operators allowing rank-one decompositions, extending finite-dimensional results to infinite dimensions.

Findings

Finite-dimensional problem reformulated as a linear program over measures.

Strong duality established for the finite-dimensional case.

Negative answer to the infinite-dimensional decomposition question.

Abstract

A problem by Feichtinger, Heil, and Larson asks whether every infinite matrix $A$ with $\sum_{k,l}|A_{kl}| < \infty$ (an equivalent substitute for the Feichtinger algebra) that is positive-semidefinite admits a symmetric rank-one decomposition $A = \sum_k f_k^*\otimes f_k$ with $\sum_k \|f_k\|_{1}^2 < \infty$. In the finite-dimensional setting, we analyze the corresponding quantitative $\ell_1^n$ optimization problem by an exact reformulation as a linear program over measures, derive its dual, and prove strong duality. We then obtain an equivalent adjoint formulation regarding the quality of a convex relaxation. In the infinite-dimensional setting, we first provide a negative answer to this question using a concurrent finite-dimensional result by Bandeira-Mixon-Steinerberger. We further study the collection of operators for which such decomposition exists, showing that they are dense in a suitable topology and invariant under the action of the positive-coefficient analytic Wiener subalgebra. In addition, we give a sufficient condition for successful rank-one decomposition in terms of $2$-summing factorization, and we characterize exactly when $A^{1/2}$ is $2$-summing.