Please use this identifier to cite or link to this item:
http://hdl.handle.net/10603/428989
Title: | Positivity preservers forbidden to operate on diagonal blocks |
Researcher: | Vishwakarma, Prateek Kumar |
Guide(s): | Khare, Apoorva |
Keywords: | Mathematics Physical Sciences |
University: | Indian Institute of Science Bangalore |
Completed Date: | 2021 |
Abstract: | The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at opposite ends , and in both cases the preservers have to be absolutely monotonic. The goal of this thesis is to complete the classification of positivity preservers that act entrywise except on specified diagonal/principal blocks , in every case other than the two above. (In fact we achieve this in a more general framework.) The ensuing analysis yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic. newline |
URI: | http://hdl.handle.net/10603/428989 |
Appears in Departments: | Mathematics |
Files in This Item:
File | Description | Size | Format | |
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01_title.pdf | Attached File | 90.26 kB | Adobe PDF | View/Open |
02_prelim pages.pdf | 208.33 kB | Adobe PDF | View/Open | |
03_abstract.pdf | 62.44 kB | Adobe PDF | View/Open | |
04_table of contents.pdf | 45.04 kB | Adobe PDF | View/Open | |
05_chapter 1.pdf | 174.05 kB | Adobe PDF | View/Open | |
06_chapter 2.pdf | 206.38 kB | Adobe PDF | View/Open | |
07_chapter 3.pdf | 242.44 kB | Adobe PDF | View/Open | |
08_annexure.pdf | 75.74 kB | Adobe PDF | View/Open | |
80_recommendation.pdf | 354.66 kB | Adobe PDF | View/Open |
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