Asian Journal of Pure and Applied Mathematics

On Bishop’s Property of n-Square Metrically Equivalent Operators

Gichunge Christopher M

Department of Mathematics and Applied Sciences, Maasai Mara University, P.O. Box 861 – 20500, Narok, Kenya.

John Matuya

Department of Mathematics and Applied Sciences, Maasai Mara University, P.O. Box 861 – 20500, Narok, Kenya.

Edward Njuguna

Department of Mathematics and Applied Sciences, Maasai Mara University, P.O. Box 861 – 20500, Narok, Kenya.

Victor Wanjala *

Department of Mathematics and Applied Sciences, Maasai Mara University, P.O. Box 861 – 20500, Narok, Kenya.

*Author to whom correspondence should be addressed.

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Abstract

We introduce a new operator equivalence relation termed as n-Square Metrically Equivalent Operators. Given a positive integer n, two bounded linear operators A and B are said to be square metrically equivalent operators if they satisfy the relation A^∗2A²ⁿ = B^∗2B²ⁿ ∀n ∈ R+. This concept generalizes the classical square-metric equivalence whenever n = 1, and allows the study of operator pairs that share deeper structural and spectral similarities. We establish that this relation forms an equivalence class and explore its key algebraic and spectral properties. We also examine how the equivalence interacts with notable operator classes, including n- square normal operators.

Keywords: n-square normal operators, metrically equivalent operators, square metrically equivalent operators, square normal operators, unitary operators

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