On symmetricity of the norm derivatives orthogonality in operator spaces

TL;DR

This paper explores the properties of $ ho$-orthogonality in operator spaces, characterizing symmetric operators in finite and infinite dimensions, and establishing conditions under which operators exhibit symmetry.

Contribution

It provides new characterizations of $ ho$-orthogonality and symmetry of operators, especially in finite and infinite-dimensional Hilbert spaces, including the unique cases in low dimensions.

Findings

In 2D real spaces, only scalar multiples of orthogonal matrices are symmetric.

In higher finite dimensions, only the zero operator is symmetric.

In infinite-dimensional spaces, the zero operator is the only symmetric operator within a broad class.

Abstract

We investigate $\rho$-orthogonality and its local symmetry in the space of bounded linear operators. A characterization of Hilbert space operators with symmetric numerical range is established in terms of $\rho$-orthogonality. Further, we provide characterizations of $\rho$-left and $\rho$-right symmetric operators on finite-dimensional Hilbert spaces. In the two-dimensional real case, we show that the only nonzero $\rho$-left (or $\rho$-right) symmetric operators are scalar multiples of orthogonal matrices. However, in any finite-dimensional Hilbert space of dimension greater than two, an operator is $\rho$-left (or $\rho$-right) symmetric if and only if it is the zero operator. For infinite-dimensional spaces, we show that within a large class of operators, the zero operator remains the only example of $\rho$-left and $\rho$-right symmetric operators.