Abstract:
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.
