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<p>We introduce the notion of local orthogonality preserving operators to study the right-symmetry of operators. As a consequence of our work, we show that any smooth compact operator defined on a smooth and reflexive Banach space is either a rank one operator or it is not right-symmetric. We show that there are no right-symmetric smooth compact operators defined on a smooth and reflexive Banach space that fails to have any non-zero left-symmetric point. We also study approximately orthogonality preserving and reversing operators (in the sense of Chmieliński and Dragomir). We show that on a finite-dimensional Banach space, an operator is approximately orthogonality reversing (preserving) in the sense of Dragomir if and only if it is close to a scalar multiple of an isometry.</p>

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On symmetric and approximately symmetric operators

  • 期刊论文

  • 发表日期：

    2025年2月3日

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  • Divya Khurana

摘要

We introduce the notion of local orthogonality preserving operators to study the right-symmetry of operators. As a consequence of our work, we show that any smooth compact operator defined on a smooth and reflexive Banach space is either a rank one operator or it is not right-symmetric. We show that there are no right-symmetric smooth compact operators defined on a smooth and reflexive Banach space that fails to have any non-zero left-symmetric point. We also study approximately orthogonality preserving and reversing operators (in the sense of Chmieliński and Dragomir). We show that on a finite-dimensional Banach space, an operator is approximately orthogonality reversing (preserving) in the sense of Dragomir if and only if it is close to a scalar multiple of an isometry.