Square-Metrically Equivalent Operators and their Closure under Unitary Equivalence

This paper introduces a new class of equivalence relation called square-metrically equivalent operators. It explores the properties of this class and its relation to other equivalence classes like unitary equivalence and metric equivalence. The study is justified by the need for new insights in operator theory, and it has practical relevance for simplifying complex operator structures. The results show that square-metric equivalence is closed under unitary equivalence and that metric equivalence implies square-metric equivalence. This research opens new pathways in operator theory and proposes new methods for handling operator-related problems