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John Matuya
Maasai Mara University
Mathematics

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On Deformed-Metric Equivalence of Hilbert Space Operators Amenya Collins; Victor Wanjala; John Matuya
Jambura Journal of Mathematics Vol 8, No 2: August 2025
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/jjom.v8i2.36685

Abstract

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 SPS = TT. 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.
Co-Authors Amenya Collins Victor Wanjala