Abstract
The paper investigates linear functionals ϕ : ℝn → ℝ preserving a set ℳ⊆ ℝ, i.e., ϕ : ℝn → ℝ such that ϕ(v) ∈ ℳ for any vector u ∈ ℝn 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 : Mn,m → Mn,m such that all entries of the matrix Φ(A) belong to ℳ for any matrix A ∈ Mn,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.