On (n,m)-Square Metrically Equivalent Operators

Abstract

We introduce and study a new equivalence relation among bounded linear operators, termed (n,m)-Square Metrically Equivalent Operators. Given positive integers n and m, two bounded linear operators A and B are said to be (n,m)-square metrically equivalent if they satisfy the relation A∗nAm = B∗nBm. This definition generalizes the classical notions of metric and square-metric equivalence, extending them to a broader framework that captures deeper algebraic and spectral similarities between operators. We show that this relation forms an equivalence class and investigate its algebraic, spectral, and structural properties. Furthermore, we explore how (n,m)-square metric equivalence interacts with well-known operator classes such as (n,m)-normal and quasi-similar operators, and we establish conditions under which important properties including Bishop’s property, isoloid, and polaroid behaviors are preserved.