The question of which functions acting entrywise preserve positive
semidefiniteness has a long history, beginning with the Schur product
[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.
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
thesis. Finally, we present the (remaining) main results in the thesis, and
conclude with some of the proofs.