On symmetric and approximately symmetric operators

TL;DR

This paper investigates the properties of symmetric and approximately symmetric operators in Banach spaces, revealing conditions under which smooth compact operators are right-symmetric or not, and characterizing approximately orthogonality preserving operators.

Contribution

It introduces local orthogonality preserving operators, characterizes right-symmetric smooth compact operators, and establishes that approximately orthogonality preserving operators are injective in finite dimensions.

Findings

Smooth compact operators are either rank one or not right-symmetric.

No right-symmetric smooth compact operators lack non-zero left-symmetric points.

Approximately orthogonality preserving operators are exactly the injective operators in finite-dimensional spaces.

Abstract

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 Chmieli\'{n}ski and Dragomir). We show that on a finite-dimensional Banach space, an operator is approximately orthogonality preserving (reversing) in the sense of Dragomir if and only if it is an injective operator.