## Positivity preservers forbidden to operate on diagonal blocks

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- by Prateek Kumar Vishwakarma PDF
- Trans. Amer. Math. Soc.
**376**(2023), 5261-5279 Request permission

## 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.

We 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.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.

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## Additional Information

**Prateek Kumar Vishwakarma**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore, India
- Address at time of publication: Department of Mathematics and Statistics, Room 307:14, College West, 3737 Wascana Parkway, University of Regina, SK S4S 0A2, Canada
- Email: prateekv@iisc.ac.in
- Received by editor(s): March 10, 2020
- Received by editor(s) in revised form: July 8, 2020
- Published electronically: May 17, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 5261-5279 - MSC (2020): Primary 15B48, 26A21; Secondary 15A24, 15A39, 15A45, 30B10
- DOI: https://doi.org/10.1090/tran/8256