Abstract
We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian matrix valued functions, with the same Poincaré constant. This generalizes recent results [ABY19, Kat20] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.
Funding Statement
Nikhil Srivastava has been supported by NSF Grant CCF-1553751.
Acknowledgments
We are grateful to the anonymous referee for helpful comments which improved the paper.
Citation
Ankit Garg. Tarun Kathuria. Nikhil Srivastava. "Scalar Poincaré implies matrix Poincaré." Electron. Commun. Probab. 26 1 - 4, 2021. https://doi.org/10.1214/21-ECP371
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