On (α, β)–Almost Similar Operators in Hilbert Spaces

This paper introduces and investigates a novel generalization of operator similarity, termed (α,β)--almost similarity, which extends the concept of almost similar operators by incorporating two real parameters. We establish fundamental properties of this new equivalence relation, demonstrating that it forms an equivalence class on the space of bounded linear operators on a Hilbert space. Key results include the invariance of spectrum, point spectrum, and approximate point spectrum under this relation. The study also defines the class of (α,β)-𝔗 operators, an expansion of the classical 𝔗-operator concept, and explores its relationship with (α,β)--almost similarity. Furthermore, we analyze the connections between similarity, unitary equivalence, and (α,β)--almost similarity, providing conditions under which these relations coincide, particularly for self-adjoint and projection operators. The results contribute to the broader understanding of operator equivalence relations and their spectral implications.