Linear operators preserving combinatorial matrix sets

P. M. Shteyner

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 181-199

Résumé

The paper investigates linear functionals $\phi : \mathbb{R}^n \rightarrow \mathbb{R}$ preserving a set $\mathcal{M} \subseteq \mathbb{R}$, i.e., $\phi : \mathbb{R}^n \rightarrow \mathbb{R}$ such that $\phi(v) \in \mathcal{M}$ for any vector $v \in \mathbb{R}^n$ with components from $\mathcal{M}$. For different types of subsets of real numbers, characterizations of linear functionals that preserve them are obtained. In particular, the sets $\mathbb{Z}, \mathbb{Q}, \mathbb{Z}_+, \mathbb{Q}_+, \mathbb{R}_+$, several infinite sets of integers, bounded and unbounded intervals, and finite subsets of real numbers are considered. A characterization of linear functionals preserving a set $\mathcal{M}$ also allows one to describe linear operators preserving matrices with entries from this set, i.e., operators $\Phi : M_{n, m} \rightarrow M_{n, m}$ such that all entries of a matrix $\Phi(A)$ belong to $\mathcal{M}$ for any matrix $A \in M_{n, m}$ with all entries in $\mathcal{M}$. As an example, linear operators preserving $(0, 1)$, $(\pm 1)$, and $(\pm 1, 0)$-matrice are characterized.