The article ‘Schur multipliers in Schatten-von Neumann classes’ signed by José Manuel Conde Alonso (ICMAT-UAM), Adrián González Pérez (ICMAT-UAM), Javier Parcet (ICMAT-CSIC) and Eduardo Tablate (ICMAT-CSIC) has been published in the journal Annals of Mathematics.

Image: Íñigo de Amescua/ICMAT

Schur multipliers are linear maps on matrix algebras with a great impact on functional analysis, operator algebras, geometric group theory and harmonic analysis.

Despite a rather simple definition, Schur multipliers have played a key role in landmark results since the mid 20th century. The celebrated Grothendieck’s inequality (1956) is intimately connected to a striking characterization of the operator boundedness of Schur multipliers. The impact of Schur multipliers in geometric group theory and operator algebras was early recognized by Haagerup. His pioneering work on free groups and the research thereafter on semisimple lattices (1979-1989) encoded deep geometric properties of these groups in terms of approximation properties for Toeplitz-like Schur multipliers on their matrix algebras.

In a different direction, Schur multipliers of divided differences were essential in Potapov/Sukochev’s celebrated solution (2011) of Krein’s conjecture on operator-Lipschitz functions. More recently, due to a close connection between Fourier and Schur multipliers, the remarkable work of Lafforgue/de la Salle unraveled  unprecedented pathologies in the Lp-convergence of Fourier series over semisimple Lie groups by analyzing Schur Sp-approximation. The Lp-theory has gained a considerable momentum since then.

Now, José Manuel Conde Alonso (ICMAT-UAM), Adrián González Pérez (ICMAT-UAM), Javier Parcet Hernández (ICMAT-CSIC) y Eduardo Tablate (ICMAT-CSIC) give an unexpected and strikingly simple criterion for the boundedness of Schur multipliers on Schatten p-classes, which solves a conjecture proposed by Mikael de la Salle and goes beyond best-known to date Sp-estimates for divided differences and Toeplitz-like Schur multipliers.

Their work is published in the journal Annals of Mathematics. The main result is a full matrix (nonToeplitz/nontrigonometric) amplification of the fundamental Hörmander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders σ>n/2 as well. This provides a one-line proof of Krein’s conjecture for Sp-multipliers and moreover extends it to α-divided differences. It also leads to strong applications in harmonic analysis for high rank simple Lie group algebras which are left to a forthcoming paper.

Reference: José M. Conde-Alonso, Adrián M. González-Pérez, Javier Parcet, Eduardo Tablate, Schur multipliers in Schatten-von Neumann classes. Annals of Mathematics Volume 198 (2023), Issue 3, Pages 1229-1260