Entrywise positivity preservers in fixed dimension: I

Abstract: Which functions preserve positive semidefiniteness (psd) when applied entrywise to the entries of psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, who classified the positivity preservers of matrices of all dimensions. The study of positivity preservers in fixed dimension is harder, and a complete characterization remains elusive to date. In fact until recent work, it was not known if there exists any analytic preserver with negative coefficients.

In my first talk, I will explain the classical history and modern motivations of this problem, followed by a “restricted” solution in every dimension. Central to the proof are novel determinantal identities involving Schur polynomials. I will conclude with a few outstanding questions.

(Based on two papers: with Alexander Belton, Dominique Guillot, and Mihai Putinar, Adv. Math. 2016; and with Terence Tao, Amer. J. Math., in press.)

functional analysis

Audience: researchers in the topic

Functional Analysis and Operator Theory Webinar

Series comments: The aim of this lecture series is to provide the community with an opportunity for regular online meetings these days when physical seminars and conferences are not possible.

This seminar is a continuation of the Preserver Webinar series with an intention to widen its scope and range of speakers. If you are interested in following the updates, please write an e-mail to faot[dot]webinar[at]gmail[dot]com with the subject "Subscribe".