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. The talk will begin by discussing connections between metric geometry and positivity, also via positive definite functions. Following this, we present Schoenberg's motivations in studying entrywise positivity preservers, followed by classical variants for matrices with entries in other real and complex domains. Then we shall see the result due to Guillot and Rajaratnam on preservers acting only on the off-diagonal entries, touching upon the modern motivation behind it. This is followed by its generalization in the paper. Finally, we present the (remaining) main results and conclude with some of the proofs.
| 11/29 | Brent Cody (VCU) | Title tba |
| 11/15 | Charles Johnson | Title tba |
| 10/25 | Eric Swartz | Title tba |
| 10/11 | Fan Ge | Title tba |
| 10/04 | Prateek Kumar Vishwakarma (U. of Regina) | Positivity preservers forbidden to operate on diagonal blocks |
| 9/27 | Chi Kwong Li | Quantum tomography problem for open quantum systems |
| 9/13 | Chi Kwong Li | Matrix problems in quantum information science (Slides) |