| dc.contributor.advisor | Khare, Apoorva | |
| dc.contributor.author | Vishwakarma, Prateek Kumar | |
| dc.date.accessioned | 2021-12-03T09:14:47Z | |
| dc.date.available | 2021-12-03T09:14:47Z | |
| dc.date.submitted | 2021 | |
| dc.identifier.uri | https://etd.iisc.ac.in/handle/2005/5537 | |
| dc.description.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. | en_US |
| dc.language.iso | en_US | en_US |
| dc.rights | I grant Indian Institute of Science the right to archive and to make available my thesis or dissertation in whole or in part in all forms of media, now hereafter known. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part
of this thesis or dissertation | en_US |
| dc.subject | Entrywise Positivity Preservers | en_US |
| dc.subject | Absolutely Monotonic Functions | en_US |
| dc.subject | Forbidden Blocks | en_US |
| dc.subject | Forbidden Principal Blocks | en_US |
| dc.subject | Matrix Analysis and Positivity | en_US |
| dc.subject.classification | Mathematics | en_US |
| dc.title | Positivity preservers forbidden to operate on diagonal blocks | en_US |
| dc.type | Thesis | en_US |
| dc.degree.name | PhD | en_US |
| dc.degree.level | Doctoral | en_US |
| dc.degree.grantor | Indian Institute of Science | en_US |
| dc.degree.discipline | Faculty of Science | en_US |
