On Unitary Quasi-Square Equivalence and Related Classes of Operators

This paper introduces and systematically investigates the notion of unitary quasi-square equivalence for bounded linear operators on Hilbert spaces. This equivalence relation, defined through the unitary equivalence of operator squares, provides a classification that preserves essential spectral and structural properties while capturing higher-order similarities not detectable through classical unitary equivalence. We establish that this relation defines a genuine equivalence relation and explore its connections with fundamental operator classes including square normal operators, -quasi-normal operators, hyponormal operators, and various isometric structures. Our main contributions include complete characterization of spectral invariants preserved under this equivalence, demonstration of preservation theorems for advanced operator classes and their -algebraic structure, applications to concrete operator families, and development of decomposition theorems revealing the canonical structure of equivalence classes. The theory developed provides a tool for operator classification with applications to invariant subspace problems, similarity theory, and the structural analysis of non-normal operators.