Linear Operators Preserving Combinatorial Matrix Sets

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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 languageEnglish
Pages (from-to)114-125
Number of pages12
JournalJournal of Mathematical Sciences
Volume262
Issue number1
DOIs
StatePublished - Mar 2022
Externally publishedYes

Bibliographical note

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

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