## Abstract

The paper investigates linear functionals ϕ : ℝ^{n} → ℝ preserving a set ℳ⊆ ℝ, i.e., ϕ : ℝ^{n} → ℝ such that ϕ(v) ∈ ℳ for any vector u ∈ ℝ^{n} with components in ℳ. For various types of subsets of real numbers, characterizations of the linear functionals that preserve them are obtained. In particular, the sets ℤ,ℚ, ℤ_{+},ℚ_{+},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 ℳ also allows one to describe the linear operators preserving matrices with entries from this set, i.e., the operators Φ : M_{n,m} → M_{n,m} such that all entries of the matrix Φ(A) belong to ℳ for any matrix A ∈ M_{n,m} with all entries in ℳ. As an example, linear operators preserving (0, 1)-, (±1)-, and (±1, 0)-matrices are characterized.

Original language | English |
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Pages (from-to) | 114-125 |

Number of pages | 12 |

Journal | Journal of Mathematical Sciences |

Volume | 262 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2022 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.