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.
