On Approximately Unitarily Equivalent Operators

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.