http://hdl.handle.net/123456789/6 Thu, 17 Sep 2026 17:18:37 GMT 2026-09-17T17:18:37Z http://hdl.handle.net/123456789/19846 On Unitary Quasi-Square Equivalence and Related Classes of Operators Mwaura M Peter, John Matuya, Victor Wanjala 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. Sun, 01 Mar 2026 00:00:00 GMT http://hdl.handle.net/123456789/19846 2026-03-01T00:00:00Z http://hdl.handle.net/123456789/19845 On Deformed-Metric Equivalence of Hilbert Space Operators Collins Amenya, Victor Wanjala, John Matuya This paper introduces and studies a novel class of equivalence relations in operator theory, called deformed-metrically equivalent operators. Two operators S and T in B(H) are said to be deformed-metrically equivalent if there exists a positive operator P such that S ∗PS = T ∗ T . This definition generalizes traditional metric equivalence by incorporating a positive deformation operator P, enabling a richer algebraic and spectral analysis. We establish several fundamental results, including the preservation of key operator classes such as normality, posinormality, and compactness under suitable commutativity conditions. Spectral inclusion relations are derived under invertibility assumptions, and the equivalence is shown to be stable under limits, tensor products, and functional calculus. Moreover, the set of all operators deformed-metrically equivalent to a given operator forms an affine space that is closed in the weak operator topology. These findings deepen the theoretical framework of operator equivalence and reveal new connections with well-studied classes such as posinormal, supraposinormal, and k-quasi n-power posinormal operators. T Sat, 01 Aug 2026 00:00:00 GMT http://hdl.handle.net/123456789/19845 2026-08-01T00:00:00Z http://hdl.handle.net/123456789/19844 On (α, β)–Almost Similar Operators in Hilbert Spaces Beatrice Obiero Adhiambo, Victor Wanjala 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. Wed, 01 Apr 2026 00:00:00 GMT http://hdl.handle.net/123456789/19844 2026-04-01T00:00:00Z http://hdl.handle.net/123456789/19843 Local spectral properties for operators satisfying A nBA n = U (t)A j U (t) ∗ Victor Wanjala In this paper, we investigate the local spectral properties of bounded linear operators A and B acting on a Hilbert space H , which satisfy the generalized operator equation A nBA n = U (t)A j U (t) ∗ , t ∈ I ⊆ R, where U (t) is a strongly continuous family of unitary operators on L . The strong continuity condition means that for every vector x ∈ L , ∥U (t)x− U (s)x∥ → 0 as t → s. We analyze how this time-dependent unitary intertwining affects the transmission of local spectral properties between A and B. In particular, we establish conditions under which certain local spectral properties of A are inherited by B, and vice versa. Several concrete examples involving shift operators, diagonal matrices, and multiplication operators on L 2 spaces are provided in Section 3 to demonstrate the applicability of the developed results and to highlight phenomena arising from the unitary perturbation parameter t. Mon, 01 Jun 2026 00:00:00 GMT http://hdl.handle.net/123456789/19843 2026-06-01T00:00:00Z