On Approximately Unitarily Equivalent Operators

Stephen Wanyonyi Luketero; Bernard Mutuku Nzimbi; Stephen Kibet Moindi

doi:https://doi.org/10.14445/22315373/IJMTT-V70I9P101

International Journal of Mathematics Trends and Technology

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Volume 70 | Issue 9 | Year 2024 | Article Id. IJMTT-V70I9P101 | DOI : https://doi.org/10.14445/22315373/IJMTT-V70I9P101

On Approximately Unitarily Equivalent Operators

Stephen Wanyonyi Luketero, Bernard Mutuku Nzimbi, Stephen Kibet Moindi

Received	Revised	Accepted	Published
22 Jul 2024	26 Aug 2024	13 Sep 2024	30 Sep 2024

Abstract

In this paper, we investigate the operator relation ∥ 𝐴 − 𝑈𝑛 ∗𝐵𝑈𝑛 ∥⟶ 0, where 𝐴, 𝐵 ∈ 𝐵(ℋ) and 𝑈𝑛 ∈ 𝐵(ℋ) is a sequence of unitary operators, known as the approximate unitary equivalence between 𝐴 and 𝐵, which is an asymptotic version of the unitary equivalence of operators. We characterize operators in this relation and investigate other closely related relations. We give and prove conditions under which approximate unitary equivalence implies or is implied by or coincides with other equivalence relations.

Keywords

Approximate unitary equivalence, Approximate similarity, Similar, rank, Rank-preserving, Metric equivalence.

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Citation :

Stephen Wanyonyi Luketero, Bernard Mutuku Nzimbi, Stephen Kibet Moindi, "On Approximately Unitarily Equivalent Operators," International Journal of Mathematics Trends and Technology (IJMTT), vol. 70, no. 9, pp. 1-7, 2024. Crossref, https://doi.org/10.14445/22315373/IJMTT-V70I9P101