Linear Preservers of Infinitely Divisible Matrices
Abstract
An infinitely divisible nonnegative matrix is an entry-wise nonnegative matrix that admits an entry-wise nonnegative mth root, with respect to usual matrix multiplication, for every positive integer m; it is called strongly infinitely divisible when it is, in addition, invertible. The study of such matrices has its origins in the theory of infinitely divisible probability distributions and the embedding problem for Markov matrices and is closely connected with continuous one-parameter semigroups; in the invertible case, this connection admits a natural description in terms of the exponential map. In this paper, we characterize the bijective linear maps on Mn(ℝ) that preserve strongly infinitely divisible matrices and infinitely divisible nonnegative matrices. In the former case, we combine the Inverse Function Theorem with Zariski-density techniques, using the Zariski-density approach recently developed in linear preserver theory by Fallat and Mondal [\textit{Proc. Amer. Math. Soc.}, 2026]. In the latter case, we first show that preserving infinite divisibility forces preservation of the cone of entry-wise nonnegative matrices and then exploit the additional structure afforded by infinite divisibility to complete the characterization.
Related
- Directional Optimal Sub-Gamma Scales for Infinitely Divisible Laws
- Risk diversification for infinitely divisible distributions
- Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws
- Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors
- On an infinitely divisible distribution involving modified Bessel functions
BibTeX
@misc{fallat2026linear,
title = {{Linear Preservers of Infinitely Divisible Matrices}},
author = {Shaun Fallat and Samir Mondal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09252},
eprint = {2609.09252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09252},
}
RIS
TY - UNPB
TI - Linear Preservers of Infinitely Divisible Matrices
AU - Shaun Fallat
AU - Samir Mondal
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09252
UR - https://arxiv.org/abs/2609.09252
AB - An infinitely divisible nonnegative matrix is an entry-wise nonnegative matrix that admits an entry-wise nonnegative mth root, with respect to usual matrix multiplication, for every positive integer m; it is called strongly infinitely divisible when it is, in addition, invertible. The study of such matrices has its origins in the theory of infinitely divisible probability distributions and the embedding problem for Markov matrices and is closely connected with continuous one-parameter semigroups; in the invertible case, this connection admits a natural description in terms of the exponential map. In this paper, we characterize the bijective linear maps on M_n(ℝ) that preserve strongly infinitely divisible matrices and infinitely divisible nonnegative matrices. In the former case, we combine the Inverse Function Theorem with Zariski-density techniques, using the Zariski-density approach recently developed in linear preserver theory by Fallat and Mondal [\textit{Proc. Amer. Math. Soc.}, 2026]. In the latter case, we first show that preserving infinite divisibility forces preservation of the cone of entry-wise nonnegative matrices and then exploit the additional structure afforded by infinite divisibility to complete the characterization.
ER -