2025/12/12 by Ghosh, Souvik, Paul, Kallol, Sain, Debmalya
Mathematics · Computer Science · #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Holomorphic and Operator Theory
paper · doi:10.48550/arxiv.2512.11208
We investigate ρ-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 ρ-orthogonality. Further, we provide characterizations of ρ-left and ρ-right symmetric operators on finite-dimensional Hilbert spaces. In the two-dimensional real case, we show that the only nonzero ρ-left (or ρ-right) symmetric operators are scalar multiples of orthogonal matrices. However, in any finite-dimensional Hilbert space of dimension greater than two, an operator is ρ-left (or ρ-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 ρ-left and ρ-right symmetric operators.