1. Positivity preservers forbidden to operate on diagonal blocks
- Author
-
Prateek Kumar Vishwakarma
- Subjects
Power series ,Applied Mathematics ,General Mathematics ,Diagonal ,Monotonic function ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Combinatorics ,Mathematics - Classical Analysis and ODEs ,Converse ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,15B48, 26A21 (primary), 15A24, 15A39, 15A45, 30B10 (secondary) ,Schur product theorem ,Mathematics - 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., Minor edits in exposition, 19 pages. The paper now uses the style file of Trans. AMS (to appear)
- Published
- 2023