2024/09/05 by Todd Kemp, Kemp, Todd, Akihiro Miyagawa +1 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Matrix Theory and Algorithms #Operator Algebras (math.OA) #Point processes and geometric inequalities #Quantum Algebra (math.QA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2409.03177
openalex publication_date 2024/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Together with Speicher, in 2007 the first author proved the strong Haagerup inequality for operator norms of homogeneous holomorphic polynomials in freely independent \mathscrR-diagonal elements (including in particular circular random variables); the inequality improved the bound from the original Haagerup inequality to grow with √(n), rather than linearly in n, on homogeneous polynomials of degree n. In this paper, we prove a similar inequality for q-circular systems for |q|<1, generalizing the free case when q=0. In particular, we prove the strong Haagerup inequality for systems exhibiting neither free independence nor \mathscrR-diagonality. As an application, we prove a strong ultracontractivity theorem for the q-Ornstein--Uhlenbeck semigroup, and prove sharp rates for the Haagerup and ultracontractive inequalities.