Left inverses of matrices with polynomial decay

2008/01/10 by Romain Tessera, Tessera, Romain
Mathematics · #47B37 #47B38 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.0801.1532

openalex publication_date 2008/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The algebra of Schur operators on l² is known not to be inverse-closed. When l²=l²(X) where X is a metric space, we can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra for a strictly polynomial weight is inverse-closed. Here, we prove a result dealing with left-invertibility. Namely, if such an operator is bounded below in l^p for some p, then it is bounded below for all q, and it admits a left-inverse in the weighted Schur algebra.

Citations

• Pseudodifferential operators and Banach algebras in mobile communications

Related

• The inclusion of the Schur algebra in B(l²) is not inverse-closed
• Inverse-closedness of weighted Schur and BGS type quasi-Banach algebras
• On the inverse-closedness of operator-valued matrices with polynomial off-diagonal decay
• Vanishing Pseudo Schur Complements, Reverse Order Laws, Absorption Laws and Inheritance Properties
• Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs