Linear Preservers of Semipositivity and Minimal Semipositivity
Abstract
We classify all real linear into preservers of semipositivity on Mm,n(ℝ), m,n≥2, for arbitrary source and target pairs of proper cones, without assuming invertibility or a prescribed form. The classification consists of product maps L(A)=XAY and degenerate maps L(A)=Ψ(uTA), subject to explicit necessary and sufficient cone conditions. For square minimal semipositivity with respect to the standard nonnegative orthant, preservation instead forces the operator to be invertible: for n≥2, the linear into preservers are exactly the maps L(A)=XAY and L(A)=XATY with X,Y invertible and X−1,Y−1≥0. We also classify all linear preservers of orthant semipositivity from M2,2(ℝ) to M1,3(ℝ) into three disjoint classes, two of which lie outside product form.
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BibTeX
@misc{tian2026linear,
title = {{Linear Preservers of Semipositivity and Minimal Semipositivity}},
author = {Yulong Tian and Jinli Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27413},
eprint = {2610.27413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27413},
}
RIS
TY - UNPB
TI - Linear Preservers of Semipositivity and Minimal Semipositivity
AU - Yulong Tian
AU - Jinli Xu
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.27413
UR - https://arxiv.org/abs/2610.27413
AB - We classify all real linear into preservers of semipositivity on M_m,n(ℝ), m,n≥2, for arbitrary source and target pairs of proper cones, without assuming invertibility or a prescribed form. The classification consists of product maps L(A)=XAY and degenerate maps L(A)=Ψ(u^TA), subject to explicit necessary and sufficient cone conditions. For square minimal semipositivity with respect to the standard nonnegative orthant, preservation instead forces the operator to be invertible: for n≥2, the linear into preservers are exactly the maps L(A)=XAY and L(A)=XA^TY with X,Y invertible and X^−1,Y^−1≥0. We also classify all linear preservers of orthant semipositivity from M_2,2(ℝ) to M_1,3(ℝ) into three disjoint classes, two of which lie outside product form.
ER -