Abstract:
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, n -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 C
*
-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.
