On Bishop’s Property of n-Square Metrically Equivalent Operators

We introduce a new operator equivalence relation termed as n-Square Metrically Equivalent Operators. Given a positive integer n, two bounded linear operators A and B are said to be square metrically equivalent operators if they satisfy the relation A∗2A2n = B∗2B2n ∀n ∈ R+. This concept generalizes the classical square-metric equivalence whenever n = 1, and allows the study of operator pairs that share deeper structural and spectral similarities. We establish that this relation forms an equivalence class and explore its key algebraic and spectral properties. We also examine how the equivalence interacts with notable operator classes, including n- square normal operators.