2026-09-21T14:23:40Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/113210 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Junge, Marius Boca, Florin Oikhberg, Timur Leditzky, Felix Li, Haojian 2022-01-12T22:35:19Z 2022-01-12T22:35:19Z 2024-01-12T22:35:30Z 2021-07-16 2021-08 In this thesis, we study Sobolev type inequalities in the noncommutative (quantum) settings. We establish the abstract Bakry-\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\'Emery theorem for operator-valued functions on Riemannian manifolds. We discuss examples including noncommutative $f$-Sobolev inequalities on the intervals, Lindblad operators of finite dimensional matrix algebras, and discrete graphs. By generalizing the monotone metrics in the space of quantum states, we develop a deeper understanding of $f$-Sobolev inequalities in the noncommutative setting. Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01 The student, Haojian Li, accepted the attached license on 2021-07-15 at 19:32. The student, Haojian Li, submitted this Dissertation for approval on 2021-07-15 at 19:37. This Dissertation was approved for publication on 2021-07-16 at 13:22. DSpace SAF Submission Ingestion Package generated from Vireo submission #16972 on 2022-01-12 at 12:55:25 Made available in DSpace on 2022-01-12T22:35:19Z (GMT). No. of bitstreams: 3 LI-DISSERTATION-2021.pdf: 518207 bytes, checksum: aa4f6c0bea05d60ac824377b450a03a0 (MD5) LICENSE.txt: 4207 bytes, checksum: d9799aa709d787bce779fa968ea5a763 (MD5) PROQUEST_LICENSE.txt: 4553 bytes, checksum: 92c4638118d7507f29d06479ae30c5e3 (MD5) Previous issue date: 2021-07-16 Embargo set by: Seth Robbins for item 121136 Lift date: 2024-01-12T22:35:30Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system U of I Only application/pdf http://hdl.handle.net/2142/113210 en Copyright 2021 Haojian Li Sobolev inequalities noncommutative analysis open quantum system Noncommutative Sobolev type inequalities text Thesis Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.